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# Pipes and Water Tank
In pipes and water tank we will learn how to solve different types of problems. A water tank or a cistern is connected with two types of pipes to fill and empty it. The pipe which fills the tanks up is called the inlet. The pipe which empties it is called the outlet.
Suppose, if an inlet fills up the cistern in 5 hours, then in 1 hour it fills up 1/5 th part of it. We say that the work done by inlet in 1 hours is 1/5.
Similarly, if an outlet empties out the cistern in 4 hours, then in 1 hour it empties out 1/4 th part of the cistern. We say that the work done by the outlet in 1 hour is 1/4.
Now we will apply the concept of solving some real-life problems on pipes and a water tank or a cistern.
Word problems on pipes and water tank or cistern:
1. A cistern can be filled by a tap in 5 hours by the other tap in 4 hours. If both taps are opened together how long will it take to fill the cistern?
Solution:
Time taken by the 1st tap to fill the cistern = 5 hours
Therefore, work done by the 1st tap in 1 hour = 1/5
Time taken by the 2nd tap to fill the cistern = 4 hours.
Therefore, work done by the 2nd tap in 1 hour = 1/4
Therefore, work done by the both taps in 1 hour = 1/4 + 1/5
= (4 + 5)/20
= 9/20
Therefore, both taps will fill the cistern in = 20/9 hours.
2. A tank can fill by the taps in 8 hours and can be emptied by the other taps in 10 hours. How long will it take to fill the tank if both the taps are opened together?
Solution:
Time taken by the 1st tap to fill the tank = 8 hours
In 1 hour, the tap fills 1/8 of the tank.
Time taken by the other tap to empty the tank = 10 hours
In 1 hour, the other tap empty -1/10 of the tank (since, empty is taken as negative)
Therefore, in 1 hour work done by tap A and tap B = 1/8 – 1/10
= (5 – 4)/ 40
= 1/40
Therefore, both the taps when opened together will fill the tank in 40 hours.
3. A tank can be filled by one tap in 4 hour and empty by an outlet pipe in 6 hours. How long will it take to fill the tank if both the tap and pipe are opened together?
Solution:
Time taken by tap to fill the tank = 4 hours
In 1 hour, the tap fill 1/4 th part of tank.
Time taken by pipe to empty the tank = 6 hours
In 1 hour, the pipe empties 1/6 th part of the tank.
Thus, in one hour (1/4 – 1/6) th = (3 – 2)/12) th
= 1/12 th part of the tank is filled.
Therefore, the tank will be filled in 12 hours.
`
Calculate Time to Complete a Work
Calculate Work Done in a Given Time
Problems on Time required to Complete a Piece a Work
Problems on Work Done in a Given Period of Time
Problems on Time and Work
Pipes and Water Tank
Problems on Pipes and Water Tank
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Listen and Read Activities
The Path of a President: A Listen and Read Book, Level B
Early readers follow Abraham Lincoln from childhood to the presidency with this short read-along that includes text, audio, and photos.
- Grades: 1–2
“The Path of a President” (grade 1) – a brief Listen & Read nonfiction story – shares Abraham Lincoln’s path from the log cabin of his childhood to the highest office in the land. Along the way, students learn where he grew up, former jobs he had, and the woman he married. This reading activity includes text, images, and audio, so students can read along while listening to a narrator.
At the end of the story, a Sound It Out section lets students listen to these vocabulary words a second time: "cabin," "listen," "school," and "store."
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Health officials are concerned about human exposure to West Nile virus from mosquitoes. Though there have been no human cases reported in Weld County this year, mosquitoes show high levels of infection.
Health officials recommend that the following precautions be taken to avoid mosquitoes around the home:
» Drain standing water outside the home, such as in pails, trash cans and pots. Use a Mosquito Dunk (larvacide) if standing water cannot be drained.
» Avoid over-watering landscapes and yards. Mosquitoes can breed even in small amounts of stagnant water.
Recommendations to prevent mosquito bites include:
» Apply mosquito repellent to exposed skin and clothing.
» Wear light colors and loose fitting clothing. Mosquitoes seem to be attracted to darker colors and can bite through tight clothing.
» Avoid the outdoors between dusk and dawn, when mosquitoes are most active.
The health department began monitoring mosquitoes in early June through mosquito traps set by Colorado mosquito control. The Culex mosquito, known to spread West Nile, is tested to determine risk to humans. Some municipalities have mosquito control programs, but cannot completely eliminate mosquitoes.
West Nile is carried by birds and transmitted by mosquitoes. There are no treatments or vaccines. Most people infected have no symptoms. About one in five will develop a fever and other symptoms. Less than 1 percent develop a serious, sometimes fatal neurologic illness. People who develop symptoms should contact their health care providers immediately.
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3.3. Simple Objects Definition 3.3.1. (a) A mono (or subobject) is called a fraction if it is coflat and normal. (b) A map to an object X is called generic if it is not disjoint with any non-initial analytic subobject of X. Proposition 3.3.2. (a) The class of fractions is closed under composition and stable under pullback. (b) If f: Y --> X is a fraction and U is a strong subobject of Y, then U --> f+1(U) is a fraction. (c) If f: Y --> X is a fraction which factors through a mono u: U --> X in a map v: Y --> U, then v is a fraction. (d) Any proper fraction of an object is disjoint with a non-initial strong subobject. Proof. (a) The classes of coflat maps and normal monos are closed under composition and stable under pullback. (b) Since f is coflat, we have f-1f+1(U) = U by (1.5.4). Thus U --> f+1(U) is the pullback of f along u: f+1(U) --> X, therefore is a fraction by (a). (c) The mono v is the pullback of f along u, thus is fractional by (a). (d) Suppose U is a proper fraction of an object X. Since U is normal, U is disjoint with a non-initial map t: T --> X, and we may assume t is a strong mono by (1.5.2). –– Definition 3.3.3. (a) A map to an object X is called local if it is not disjoint with any non-initial strong subobject of X; a non-initial object is called pseudo-simple if any non-initial map to it is local. (c) A map to an object X is called quasi-local if it does not factor through any proper fraction to X; a non-initial object is called quasi-simple if any non-initial map to it is quasi-local. (d) A map to an object X is called prelocal if it does not factor through any proper analytic mono to X; a non-initial object is called presimple if any non-initial map to it is prelocal. Proposition 3.3.4. (a) Any local map is quasi-local; any quasi-local map is prelocal. (b) The class of local (resp. generic, resp. quasi-local, resp. prelocal) maps is closed under composition. (c) A quasi-local fraction (resp. prelocal analytic mono) is an isomorphism. (d) If f: Y --> X and g: Z --> Y are two maps and gf is local (resp. generic, resp. quasi-local, resp. prelocal) then f is local (resp. generic, resp. quasi-local, resp. prelocal). (e) Any unipotent map is both local and generic. (f) Any epi is generic. (g) If f: Y --> X is a generic map and Y is quasi-primary then X is quasi-primary. Proof. (a) Any proper fraction u: U --> X is disjoint with a non-initial strong subobject V of X by (3.3.2.d). Any local map f: Y --> X is not disjoint with V, therefore f does not factor through any proper fraction U. The second assertion is trivial. (b) Consider two local maps f: Y --> X and g: Z --> Y. Suppose fg is disjoint with a strong mono v: V --> X. Then f-1(V) --> Y is disjoint with g: Since g: Z --> Y is local this implies that f-1(V) is initial. Thus f is disjoint with V. Since f is local, V is initial. This shows that fg is local. The proof for generic maps is similar. Consider two quasi-local maps f: Y --> X and g: Z --> Y. Suppose f°g factors through a fraction v: V --> X. Then g factors through f-1(V). Since g is quasi-local and f-1(V) --> Y is a fraction, we have f-1(V) = Y. Thus f factors through V, so V = X as f is quasi-local. This shows that gf is quasi-local. The proof for prelocal maps is similar. (c) - (e) are obvious. (f) If f: Y --> X is an epi then its pullback along any non-initial analytic mono V --> X is a non-initial epi W --> V, thus f is not disjoint with V --> X. Hence f is generic. (g) Consider two non-initial analytic subobjects U and V of X. Since f is generic, the analytic subobjects f-1(U) and f-1(V) of Y are non-initial. Since Y is quasi-primary, f-1(V) f-1(U) is non-initial. Since the induced map f-1(V) f-1(U) --> X of f factors through U V, it follows that U V is non-initial. Thus X is quasi-primary. Recall that an epi e is called extremal provided that it does not factors through a proper mono. Every regular epi is an extremal epi. Definition 3.3.5. A non-initial object is called simple (resp. extremal simple, resp. unisimple) if any non-initial map to it is epic (resp. extremal epic, resp. unipotent). Proposition 3.3.6. An object X is simple (resp. extremal simple, resp. unisimple, resp. quasi-simple, resp. presimple) iff it has exactly two strong subobjects (resp. subobjects, resp. normal sieves, resp. fractions, resp. analytic subobjects). Proof. A non-initial strong subobject V of a simple object is determined by an epic strong mono, which must be an isomorphism by (1.1.2.b), thus V = X. Conversely, assume X has exactly two strong subobjects; then X is non-initial. If t: T --> X is any non-initial map then t+1(T) is a non-initial strong subobject of X, thus t+1(T) = X, so t is epic. A non-initial subobject V of an extremal simple object X is an extremal epic mono, so V = X. Conversely if X has exactly two subobjects, any non-initial map t: T --> X must be extremal epic. Any non-initial map to a unisimple object is unipotent, thus generates the maximal normal sieve. This implies that X has exactly two normal sieves. Conversely if X has exactly two normal sieves, any non-initial map to X generates the maximal normal sieve, thus it is unipotent. A non-initial fraction to a quasi--simple object is a quasi-local fraction, thus is an isomorphism by (3.3.4.c). Conversely, if X has exactly two fractions then any non-initial map is quasi-local, so it is quasi-simple. The proof for the case of presimple objects is similar. Proposition 3.3.7. (a) Any simple object is integral. (b) Any extremal simple object and any reduced unisimple object is simple. Proof. (a) Any simple object is reduced and primary (cf (3.3.6), (3.1.2.a) and (3.2.1.a)). (b) Clearly any extremal simple object is simple by (3.3.6). Any non-initial map f to an object X is unipotent if X is unisimple, and is epic if furthermore X is reduced. Thus any reduced unisimple object is simple. Proposition 3.3.8. (a) A non-initial object is pseudo-simple iff any non-initial strong subobject is unipotent. (b) Any simple object, extremal simple object, and unisimple object is pseudo-simple. (c) Any pseudo-simple object is quasi-simple. (d) Any quasi-simple object is presimple. (e) Any presimple object is primary. Proof. (a) By definition a non-initial object X is quasi-simple iff any non-initial map to it is not disjoint from any non-initial strong mono. This implies that any non-initial strong mono is unipotent. (b) Any non-initial strong mono to a simple or extremal simple object is an isomorphism. Any non-initial strong mono to a unisimple object is unipotent. By (a) these objects are pseudo-simple. (c) and (d) follow from (3.3.4.a) (e) Any non-initial analytic mono to a presimple object is an isomorphism, thus any presimple object is primary. Proposition 3.3.9. (a) Any reduced pseudo-simple object is simple. (b) Any non-initial strong subobject of a pseudo-simple object is pseudo-simple. (c) The radical of any pseudo-simple object is simple. Proof. (a) Suppose X is a reduced pseudo-simple object. By (3.3.8.a) any non-initial strong mono to X is unipotent, therefore is epic, thus is an isomorphism. It follows that X is simple by (3.3.6). (b) Suppose V is a non-initial strong subobject of a pseudo-simple object X . Then any non-initial strong subobject W of V is also a strong subobject of X , thus W is a unipotent subobject of X. It follows that W is also a unipotent subobject of V. Thus V is pseudo-simple by (3.3.8.a). (c) follows from (a) and (b). Proposition 3.3.10. Suppose A is locally disjunctable in which each object has a radical. The following are equivalent for an object X : (a) X is pseudo-simple. (b) X is quasi-simple. (c) X is presimple. (d) The radical of X is simple. Proof. (a) implies (b) and (b) implies (c) by (3.3.8). Assume X is presimple. Any proper strong subobject V of its radical rad(X) is not unipotent in X, thus it is disjoint with a non-initial analytic mono u: U --> X to X by (3.1.10). Since X is presimple, U = X by (3.3.6). Thus V is initial. This shows that rad(X) is simple. Thus (c) implies (d). Finally assume X is an object whose radical rad(X) is simple. We prove that it is pseudo-simple. Suppose V is a non-initial strong subobject of X. Its radical rad(V) is a non-initial strong subobject of rad(X), therefore we have rad(V) = rad(X) as by assumption rad(X) is simple. Since rad(X) is unipotent, V is unipotent. Thus X is pseudo-simple by (3.3.8.a). Proposition 3.3.11. Suppose any coflat unipotent map in A is regular epic and any map to a simple object is coflat. Then (a) Any coflat mono is normal. (b) Any simple object is extremal simple and unisimple. Proof. (a) Consider a coflat mono u: U --> X. To see that u is normal it suffices to prove that any pullback of u is not proper unipotent. Consider the pullback v: V --> Y of u along a map f: Y --> X. If v is unipotent then the coflat unipotent mono v is a regular epi by assumption, so is an isomorphism because any regular epic mono is isomorphic. (b) Consider a simple object P. We first show that P is unisimple. Consider two non-initial maps t: T --> P and s: S --> P. By assumption s and t are coflat epic. Thus the pullback of (s, t) is non-initial. This show that P is unisimple. Next we prove that P is extremal simple. Any non-initial subobject V of P is coflat unipotent, thus is a regular epi. Again since any regular epic mono is isomorphic, we have V = P as desired. Example 3.3.11.1. The category of affine schemes satisfies the conditions of (3.1.11). Thus any coflat mono of affine schemes is normal and any simple object (i.e. the spectrum of a field) is extremal simple and unisimple. [Next Section][Content][References][Notations][Home]
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# Video: Skip Counting By 100s within 1000
Start at 565 and skip count by 100s. What number will come next?
02:07
### Video Transcript
Start at 565 and skip count by 100s. What number will come next?
This question tests our ability to count. Instead of starting at zero, we’re told to start counting at 565. And instead of saying every single number, we’re told to skip count by 100s. This is the same as adding 100 every time and not saying any of the numbers in between. Let’s think about the number 565 to start with. It’s made up of five hundreds, six tens, and also five ones.
Now, as we’ve said already, by skip counting by 100s, we’re adding 100 more each time. That’s one more hundred but no more tens and no more ones. The tens and the ones digits are going to stay the same. This means that as we skip count in 100s, as we say each number, it’s going to be a number of 100s and 65. So to begin with, we have five 100s and 65. And the question asks us, what number will come next? If we add 100 to 565, we get 665.
And although we don’t have to in this question, we could carry on, adding 100 each time and skip counting by 100s. 765, 865, and so on. We could keep going. But we don’t need to go as far as this. As we’ve seen already, if we start at 565 and skip count by 100s, the number that will come next is 665. 565 plus 100 equals 665.
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Read Aloud, Read Along, Read Appropriately to Foster Flexible Readers
By: Robin Fogarty
Based on the framework of the National Assessment of Educational Progress (NAEP), there are three types of reading that dictate student proficiency: narratives (for literary experiences), informational reading (for facts, data, and a knowledge base), and procedural reading (for following directions and understanding technical works).
Further suggested in this model are four levels of reading: initial understanding, interpretation, developing a personal response, and evaluating. To help students become efficient and flexible readers moving through the different levels of proficiency, the three strategies of read aloud, read along, and read appropriately play different roles.
Reading aloud gives students the opportunity to hear the sound and rhythm of the language. As the teacher thinks aloud about what he or she is reading, the students begin to understand the connections between the words on the page and what they mean. When students read orally, they, too, can hear the words as they process them.
In the read along strategy, teachers provide needed word prompts and cues, as well as fluency in the reading act. As students follow along, their pacing is propelled by the fluency of the reader. The read along activity is a reading exercise for the classroom and for the home. Parents/guardians and older siblings can read orally as the student reads along.
Surprisingly, the fluent reader can read along at quite a brisk pace,and the student somehow seems to keep up, carried along by the flow of the oral reading. (When read-alongs arc employed as a strategy for fluency, do not point to the words, but rather place a paper marker beneath the line being read).
The read appropriately strategy promotes the policy of reading material at an appropriate instructional level for greatest individual gains. The adage "different strokes for different folks" applies well here. Because readers respond differently to the reading and writing process, their skill level is critical to their developmental progress. Read aloud, read along, and read appropriately are a triad of strategies that effectively achieve the results teachers want.
Discuss the NAEP framework for types of reading and levels of comprehension to inform students of the various types of reading. In this way, teachers expose learners to the idea of flexible reading for different purposes. Included in the types of reading are narratives for literary experiences, informative texts for Information gathering, and procedural steps for following directions. Demonstrate each type. Then discuss or think about the levels of reading comprehension: initial understanding, interpretation, personal response, and evaluation. Develop a rubric with students that helps them begin to assess their own levels of understanding about their reading (see figure below). Discuss the differences and make them aware of the ultimate goal — achieving a deep understanding of the reading.
Read aloud, read along, and read appropriately strategies involve three phases of reading instruction.
Story Time: Using juvenile literature books to highlight the idea of being literate is a powerful strategy for youngsters to use as they read to younger students. Several of the best books include Leo the Late Bloomer by Robert Kraus, Thank You, Mr. Falker by Patricia Polacco, and The Jolly Postman by Janet and Allan Ahlberg.
Readers' Theater: This read aloud approach calls on a group of students to take on roles to read, as they perform a dramatic reading. It creates a natural flow to reading aloud, set apart from the usual round robin reading. In this case, students read when their roles appear in the text. It seems to be highly motivating and engaging to students of various ages.
|Uses analogies||Reads critically|
Echo Reading: When students read side by side and echo read, they basically read in tandem. This affords students the advantage of seeing, saying, and hearing the words and the sound of the language as they read.
Partner Reading: By alternating paragraphs, sections. or pages, students have a companion reader to spark their read-alongs.
The Rule of Five: When students count (on their fingers) five unknown words from one page, the book is probably too difficult. It is called "the rule of five." but it is a simple, folksy assessment for finding an appropriate level book.
Fogarty, R.J. (2007). Literacy matters: Strategies every teacher can use. Thousand Oaks, CA: Corwin Press.
Used with permission from Corwin Press.
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## 200 Centimeter (cm) equals 78.7402 inch (in)
The 200 centimeter to inches converter is a size converter indigenous one unit to another. One centimeter is approximately 0.3937 inches.
You are watching: 200 cm is how many inches
The devices of length must be convert from centimeters to inches. The 200 centimeter to inches is the most an easy unit conversion you will learn in primary school school. This is among the most usual operations in a wide variety of mathematical applications.
This write-up explains exactly how to convert 200 cm to inches and also use the device for converting one unit indigenous another, as well as the relationship in between centimeters and inches with detailed explanations.
## Why change the length from 200 centimeter to inches to inches?
A centimeter (or centimeter) is a unit that length. It is one hundredth the a meter. However, the unified States offers a typical unit the length. Imperial units are supplied in the same means in great Britain.
The common Imperial or us unit of measure up for size (or distance) is inches. If you have actually information about length in centimeters; and also you require the exact same number in tantamount inch units, you can use this converter.
### The relationship in between inches and cm
To convert 200 centimeters come inches or inches to centimeters, the relationship in between inches and also centimeters is that one customs in the metric mechanism is exactly 2.54 centimeters.
1 customs = 2.54 cm
Therefore,
1 cm = 1 / 2.54 inch
To transform centimeters come inches, we need to divide the worth in centimeters through 2.54.
If the unit length is 1 cm, the corresponding length in inches is 1 cm = 0.393701 inches
## How many inches is 200cm
### Convert 200 cm (centimeters) to inches (in)
With this length converter we have the right to easily convert cm come inches prefer 10 cm to inches, 16 centimeter to inches, 200 cm to inches, 200cm in inches etc.
Since we understand that a centimeter is around 0.393701 inches, the conversion indigenous one centimeter come inches is easy. To convert centimeters come inches, main point the centimeter value provided by 0.393701.
For example, to transform 10 centimeters come inches, multiply 10 centimeters by 0.393701 to get the value per inch.
(i.e.) 10 x 0.393701 = 3.93701 inches.
Therefore, 10 centimeters is same to 3.93701 inches.
Now consider an additional example: 200cm in inches is converted as follows:
## How carry out I convert 200 centimeter to inches?
To convert 200 cm to in, merely take the actual measurement in cm and also multiply this number by 2. 20054. So girlfriend can transform how plenty of inches is 200 cm manually.
You can additionally easily convert centimeters come inches using the adhering to centimeters come inches conversion:
### How numerous inches is 200 cm
As us know, 1 centimeter = 0.393701 inches
## What is 200 centimeter in inches
In this way, 200 centimeters have the right to be convert to customs by multiplying 200 through 0.393701 inches.
(i.e.) 200 centimeter to one inch = 200 x 0.393701 inches
200 cm = inches = 78.7402 inches
### 200 cm is how numerous inches
Therefore, 200 cm is how countless inches 200 centimeter is equal to 11,811 inches.
## Example of converting centimeters come inches
The following instances will aid you understand exactly how to transform centimeters come inches.
### Convert 200 cm to inches
Reply:We understand that 1 centimeter = 0.393701 inches.
See more: Karl Marx Asserted That Culture: The Missing Concept, Sociology Quiz 3 Flashcards
To transform 200 centimeters to inches, main point 200 centimeters through 0.393701 inches.
= 200 x 0.393701 inches
= 78.7402 inches
200 centimeter is same to how plenty of inchesHow numerous inches is 200 centimeter equal to200 come 44 centimeter is how plenty of inchesWhat is 200 centimeter equal come in inches?Convert 200 cm to inches200 x 200 x 25 cm to inches200 cm convert to inches
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999
## What is the largest number you can write by using only 3 digits it’s not 999?
Answer: Biggest numbers that can be formed using 3 digits are as follows: If repetition of all the 3 digits is allowed, then the required number is 999. If repetition of only 2 digits is allowed, then it is 998. If no repetition is permitted, the number is 987.
900 numbers
## What is the greatest number of 3 digits which when divided by 6 9 12 leaves a remainder 3 in each case?
Answer: Required Number is 975. Step-by-step explanation: Given: Required Number when divided by 3 , 9 and 12 leaves 3 as remainder in each case.
## When the largest 3-digit number is divided by 3 we get?
We know the divisibility rule for 3: If the sum of the digits of a number is divisible by 3, then the number is also divisible by 3. Here, 9+9+6=24 is multiple of 3. So, 996 is the largest number divisible by 3.
## What is the greatest number of 4 digits that when divided by any of the number 6 9 12 17?
The Largest number of four digits is 9999.by dividing 9999 to 612 get 207 as remainder. So, the greatest number of 4 digits divisible by any numbers from 6, 9, 12 or 17 = (9999-207) = 9792.
## What is the least number when divided by 36 24 and 16?
Explanation: For 16, 24 and 36 the smallest number which would be perfectly divisible by them is their LCM which is 144.
9996
## What 4 digit numbers are divisible by 7?
The first 4-digit number divisible by 7 is 1001. This is sometimes also referred to as the smallest four digit number divisible by 7 or the lowest 4-digit number divisible by 7. What is the last four digit number divisible by 7? The last 4-digit number divisible by 7 is 9996.
## What is the greatest 4 digit number divisible by 3?
The last 4-digit number divisible by 3 is 9999. This is sometimes also referred to as the largest four digit number divisible by 3 or the greatest 4-digit number divisible by 3.
9940
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# Definition:Simultaneous Equations
## Definition
A system of simultaneous equations is a set of equations:
$\forall i \in \set {1, 2, \ldots, m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$
That is:
$\displaystyle \beta_1$ $=$ $\displaystyle \map {f_1} {x_1, x_2, \ldots x_n}$ $\displaystyle \beta_2$ $=$ $\displaystyle \map {f_2} {x_1, x_2, \ldots x_n}$ $\displaystyle$ $\cdots$ $\displaystyle$ $\displaystyle \beta_m$ $=$ $\displaystyle \map {f_m} {x_1, x_2, \ldots x_n}$
### Linear Equations
A system of simultaneous linear equations is a set of equations:
$\displaystyle \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
That is:
$\displaystyle \beta_1$ $=$ $\displaystyle \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n$ $\displaystyle \beta_2$ $=$ $\displaystyle \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n$ $\displaystyle$ $\cdots$ $\displaystyle$ $\displaystyle \beta_m$ $=$ $\displaystyle \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n$
## Solution
An ordered $n$-tuple $\tuple {x_1, x_2, \ldots, x_n}$ which satisfies each of the equations in a system of $m$ simultaneous equations in $n$ variables is called a solution of the system.
### Solution Set
$\mathbb S := \forall i \in \set {1, 2, \ldots, m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$
Let $\mathbb X$ be the set of ordered $n$-tuples:
$\set {\sequence {x_j}_{j \mathop \in \set {1, 2, \ldots, n} }: \forall i \in \set {1, 2, \ldots, m}: \map {f_i} {\sequence {x_j} } = \beta_i}$
which satisfies each of the equations in $\mathbb S$.
Then $\mathbb X$ is called the solution set of $\mathbb S$.
## Consistency
$\forall i \in \set {1, 2, \ldots m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$
that has at least one solution is consistent.
If a system has no solutions, it is inconsistent.
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Can I Become a Writer? - Summer 2In this section...
We focus on 'Elmer the Elephant' thinking about feelings.
We relate the different colours of Elmer to our own experiences and feelings. We talk about a different feeling each day and relate it to a different colour and come dressed in each day's colour.
Everyday we have a circle-time and tell our friends what makes us feel happy, sad, angry, calm and excited. We learn that it is okay to feel all of these emotions and how to ask our friends or grown ups for help if we have sad or angry feelings.
All week we play with different coloured malleable materials like, play dough, foam, sand and cornflour. We take part in some yoga and colour in special mandala patterns to help us feel calm. We also make some big patchwork Elmer cushions to use in our book corner to remind us of the different feelings we have been learning about.
We draw pictures on a square of fabric with special pens and paints to show the things that make us feel happy.
We help Mrs Woods to use a sewing machine to sew all of our patchwork squares together and make an amazing patchwork picnic blanket of all the things that make Nursery happy.
On Friday we make colourful cakes and hold an ‘Elmer Picnic Party’ on the field. What a lovely, exciting and happy end to a lovely week full of learning about emotions and being creative.
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Stone castles were an evolution of the early ‘Motte and Bailey’ castle design.
The simple Motte and Bailey was brought to England by the Normans in 1066, and the design consisted of a defensive mound with buildings on the flat-top.
The Motte and Bailey was quick to construct, but was generally made of wood. This made it extremely vulnerable to fire-flinging attacks – and quite temporary, too.
Consequently, as Norman control over England became more secure, new stone castles began to be built, or wooden Motte and Bailey castles were rebuilt in stone.
A Stone Castle Building Spree
The first – and most famous – stone castle was the White Tower of the Tower of London.
This stone tower was begun in 1070, and marked the start of a stone-castle building spree. By the time William the Conqueror died in 1087, 86 of these had been built in the UK.
Castles made of stone continued to be built (alongside traditional, timber Motte and Bailey castles) throughout the 1100s.
However, by the late 1100s, traditional wooden Motte and Baileys had firmly fallen from fashion, as nobles sought to demonstrate their influence from within mighty stone buildings.
What was special about Stone Castles?
These castles were an evolution of the Motte and Bailey design. They tended to be more defensive, more permanent, and more grand than their predecessors.
Generally, they were built of sandstone or limestone, but the whole castle wouldn’t have been made of stone – it was expensive and unwieldy. Costs would have been cut by using wooden roofs, partitions, and supports.
At a glance: advantages of Stone Castles
- Could survive attacks using fire – although some elements (such as roofs) were made of wood
- Stone walls and towers were much stronger against catapults and siege engines (although they certainly weren’t impenetrable!)
- Stone buildings would last for centuries, whereas wood lasted just years
- Stone buildings could be much larger and grander simple wooden designs – befitting the important nobles who lived in them.
The first innovation in the development of castles made of stone was the central castle tower of stone – also known as the ‘castle Keep’ or the ‘Donjon’.
Don’t confuse the word ‘donjon’ with ‘dungeon’ – it just means a rectangular tower, and the word comes from a French term. This stone tower tended to sit at the highest point of the castle. It would usually have been the wooden tower of a Motte and Bailey castle which was re-rendered in stone.
The stone Keep satisfied two purposes. Firstly, it provided much more luxurious accommodation for nobles. Their chambers could be larger, and better protected from the rain, and from the cold.
In addition (and for obvious reasons!) stone Keeps could include grand fireplaces for heat and for comfort – whereas old timber buildings would have been restricted to much smaller fires.
The second function of the stone Keep was for defence. Tall stone Keeps provided an excellent viewpoint for archers defending the castle.
Additionally, stone Keeps were almost always taller than the wooden-climbing frames that pillagers would wheel to castles. This meant that besiegers’ intentions to climb moveable wooden scaffolding onto the top of a tower would be thwarted.
In addition to the fearsome size of the stone Keep, the sheer thickness of the stone walls served as defence against missiles and weapons pelted at the castle.
The Keep would have been at the heart of any stone castle – and would be likely to have been the first part that was built. Radiating out from the Keep, you’d discover the other crucial parts of the castle.
Many castles included numerous domestic buildings upon the Bailey. These would have included kitchens, butteries, great halls and quarters for the domestic workers – all dependent on the size of the castle, of course! Discover more about the different rooms and areas of a Medieval castle.
It’s important to emphasise that each stone castle was truly unique, and often took advantage of its particular site – some castles perched upon rocky outcrops (such as Goodrich Castle in England); and the sides of other castles were protected by sheer cliffs – such as Dunnottar Castle in Scotland.
Nowadays, the only remnants of many Motte and Bailey castles are mounds of earth that you’d probably mistake to be hills!
The most important defensive aspect of the stone castle would have been the curtain wall. This wall wrapped around the entire castle, and enclosed both the Keep and the domestic buildings.
It may have been studded with arrow-holes for archers to shoot their bows, but the fundamental purpose was it was thick, heavy, and was intended to keep intruders out.
The curtain walls of many castles could be thicker than 1.5 metres, and would often be solid all the way through (as rubble at the centre of the walls made them harder to demolish).
At a glance: disadvantages of Stone Castles
- Extremely expensive – by the late 1100s, only the King and the richest nobles could afford to build these castles
- Time consuming to build – a medium sized castle would have taken a minimum of five years to build, more like 10. It was a phenomenal project to undertake
- Expensive to maintain – large, cold and frequently leaky, these castles were a burden to look after
- Designs quickly became vulnerable to attack. Every innovation (ie, a rectangular Keep) was soon overcome by attackers (in this case, they burrowed under the corners to collapse the tower)
- Financial ruin if the castle was lost. Should a castle be destroyed, the noble owner would probably have been financially ruined for the rest of his life.
Evidently, despite the perceived advantages of stone castle designs, there were some pretty significant drawbacks to this style of castle construction.
However, this didn’t do anything to dampen their desirability. The sheer expense and cost of these castles meant that those able to afford them would command huge respect in Medieval society.
How much did it cost to build Castles of Stone?
Stone castles were extremely expensive to build. This was due to the cost of the raw materials, and the amount of labour involved in the construction.
Whereas simple Motte and Bailey mounds could be constructed in a matter of weeks or months, it took years or even decades to complete a castle made of stone. In addition to manpower costs, the price of the stone itself was significant. Stone was a relatively expensive material due to mining and transportation costs.
To give you context on the amount of money spent on castles made from stone, you should consider that King Henry II, in the 1200s, had an annual income from England of about £20,000 a year.
In the 1250s, he spent £6,440 on refurbishing Dover Castle (not even building it afresh) – about a third of his annual income!
However, Dover was an exceptionally important castle. Smaller sites, like the castle tower at Newcastle, seemed to cost around £1,000 to build. It’s estimated that a standard size castle of stone, in around 1200, would have cost about £7,500 to build over five to 10 years.
However, this gives you an indication of just how expensive these castles were – one castle was equivalent to about 40% of the King’s annual income. No wonder most nobles couldn’t afford to build their own large castles!
Where are the best remaining examples of Stone Castles today?
Everywhere! You’ll be spoiled for choice in finding impressive stone castles.
Some ‘best in the bunch’ picks include:
- The red sandstone of Goodrich Castle in England;
- The mighty bulk of Dover Castle, ‘defender of England’;
- The dramatic cliff-top setting of Dunnottar, Scotland.
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How to write the word?
So that the written text is not filled with errors, it is important to check the spelling. There are several ways to do this. For example, indicative is Word, which underlines red words that are written incorrectly. It is enough to right-click on the controversial word so that in the window that opens, see the spellings of the word. However, not everyone likes to check spelling in this way.
If you want to be confident in the literacy of your sentences, you can search the net for how to write the word correctly. To do this, there are resources that allow you to check spelling. For example, you can simply copy a word or text into a window, and in a second the program will give you your text with underlined words that are written incorrectly. Similarly, online text can be checked and.
If you doubt the spelling of a single specific word, it’s best to see how it is spelled correctly here. This is the so-called online spelling dictionary, which includes dictionaries from several sources at once.
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When your left-handed child first starts writing, they may find
it easier to form certain letters of the alphabet their own special
way, and you can make writing easier for them by showing them
The table below shows the directions in which to form letters
for joined up (or "cursive" writing). You will see that some
letters have a second, alternative direction which left-handers
often find more comfortable. Encourage your child to try both and
see which they prefer.
For all children, correct letter formation from the very
beginning is the foundation for good handwriting, which is why
joined up writing will now be taught in U.K. schools to children
from age 4 onwards.
Forming letters Aa to Gg
Forming letters Mm to Zz
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1. ## Permutations/Combinations
Determine the number of ways that the 12 members of the boys' basketball team can be lined up if Joe, Taner and Josh must all be together?
10! x 2
The answer they have is 604800, which is derived from 10!/3!
How is this a permutations with identical terms? No where in this question does it say that Joe, Taner, and Josh are triplets. What's wrong with my thinking?
2. I'm getting a different answer.
Treat the 3 people who must be together as 1 person. Then there are 10! ways to arrange them. Within this group of 3, there are 3! ways for them to be arranged. so I get 10! x 3!.
3. EDIT:
I meant that I used 10! x 3! also, but they have 10!/3!
4. Hello, skeske1234!
I don't agree with their answer.
Determine the number of ways that the 12 members of the boys' basketball team
can be lined up if $\displaystyle A, B\text{ and }C$ must all be together?
The answer they have is 604,800, which is derived from 10!/3! ??
Duct-tape $\displaystyle A,B,C$ together.
. . Note that there are $\displaystyle 3! = 6$ ways they could be ordered.
Then we have 10 "people" to arrange: .$\displaystyle \boxed{ABC}\;D\;E\;F\;G\;H\;I\;J\;K\;L$
. . and there are $\displaystyle 10!$ ways.
Therefore, there are: .$\displaystyle 10! \times 6$ possible line-ups.
5. Oh man I was sitting here for like 10 minutes trying to figure out just how the hell they managed to get that. I had the exact same thought, "Why the heck would you divide by the permutation of the boys and not multiply!"
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Museum Collections - Ancient Peru - Chavín
Chavín de Huantar is located in the north-central highlands of Peru, midway between the coast and the tropical lowlands. Founded by 800 B.C., this ceremonial centre was thought of as the home of an oracular deity. The cult of Chavín was the first art style to spread over much of the Andes. Because of their portability, textiles were used to transmit the Chavín religious ideas in a graphic way throughout much of the Andes.
The site of Chavín consists of the Old Temple, a U-shaped platform which encloses a round sunken courtyard. Inside the temple is a stone idol, carved in bas-relief (sculptural carving where image is raised slightly from background) known as the Lanzón (see left). Measuring 4.5m high and embedded in the floor, this sculpture features all the elements of the distinctive Chavín style. It has feline fangs, round eyes with pendant irises, an upturned snarling mouth, and claws and talons. Other elements of Chavín imagery include jaguars, snakes and other animal-human composites, inspired by the tropical forests to the east.
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# How to use the factor theorem on $a(b^2-c^2) + b(c^2-a^2) + c(a^2-b^2)$?
I know the factor theorem i.e,
Let $P(x)$ be a polynomial of degree greater than or equal to $1$ and $a$ be a real number such that $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$.
I have an question in my textbook which is -
• Using Factor theorem , show that $a-b,b-c$ and $c-a$ are the factors of $$a(b^2 -c^2)+b(c^2-a^2)+c(a^2-b^2).$$
I can not see any polynomial over here . How can I solve the problem ?
Hints are welcome
• Replace $a$ with $X$. Find a factor. Replace $b$ with $X$. Find a factor. Replace ... Jul 10, 2015 at 10:52
Instead of a polynomial in $x$ or $y$ as usual, consider the expression $a(b^2 -c^2)+b(c^2-a^2)+c(a^2-b^2)$ to be firstly a polynomial in $a$, then in $b$ then $c$. Each use of the factor theorem on each case should give you one solution.
Hint: Using the difference of two squares we can factorise your expression as $$a(b^2 -c^2)+b(c^2-a^2)+c(a^2-b^2) = a(b-c)(b+c) + b(c-a)(c+a) + c(a-b)(a+b)$$
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# If log 12 (3)=a and log 12 (5)=b, what is log 15 (20)=?
Asked on by tarja19
giorgiana1976 | College Teacher | (Level 3) Valedictorian
Posted on
We notice that if we'll add log 12 (3)=a and log 12 (5)=b, we'll get:
log 12 (3) + log 12 (5) = a + b (1)
Since the bases are matching, we'll apply the rule of product:
log 12 (3) + log 12 (5) = log 12 (3*5)
log 12 (3) + log 12 (5) = log 12 (15) (2)
We'll substitute (1) in (2):
a + b = log 12 (15)
But log 12 (15) = 1/log 15 (12)
1/log 15 (12) = a + b
log 15 (12) = 1/(a+b) (3)
log 15 (12) = log 15 (4*3)
log 15 (4*3) = log 15 (4) + log 15 (3)
log 15 (4) = log 15 (12) - log 15 (3) (*)
Now, we'll calculate log 15 (20):
log 15 (20) = log 15 (4*5)
log 15 (4*5) = log 15 (4) + log 15 (5)
log 15 (4) = log 15 (20) - log 15 (5) (**)
We'll write log 15 (3) with respect to log 12 (3):
log 15 (3) = log 12 (3)*log 15 (12)
log 15 (3) = a*1/(a+b)
We'll write log 15 (5) with respect to log 12 (5):
log 15 (5) = log 12 (5)*log 15 (12)
log 15 (5) = b*1/(a+b)
We'll put (*) = (**):
log 15 (12) - log 15 (3) = log 15 (20) - log 15 (5)
We'll add log 15 (5) both sides:
log 15 (20) = log 15 (12) - log 15 (3) + log 15 (5)
log 15 (20) = (1+b-a)/(a+b)
We’ve answered 317,766 questions. We can answer yours, too.
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# Quick Answer: What Is The Simplest Form Of 24 60?
## What is the simplest form of 24?
What is 24/100 Simplified.
– 6/25 is the simplified fraction for 24/100..
## What percentage is 25 out of 60?
41.67Percentage Calculator: 25 is what percent of 60? = 41.67.
## How do you simplify 45 60?
Thus, 3/4 is the simplified fraction for 45/60 by using the GCD or HCF method. Thus, 3/4 is the simplified fraction for 45/60 by using the prime factorization method.
## What is the ratio of 5 to 9?
Simplify Ratio 5:9. Here we will simplify the ratio 5:9 for you and show you how we did it. To simplify the ratio 5:9, we find the greatest common divisor of 5 and 9, and then we divide 5 and 9 by the greatest common divisor.
## What is the simplest form of 15 60?
What is 15/60 Simplified? – 1/4 is the simplified fraction for 15/60. Simplify 15/60 to the simplest form.
## What simplified 32 60?
What is 32/60 Simplified? – 8/15 is the simplified fraction for 32/60.
## How do you express ratios in simplest form?
How to Simplify a Ratio A : B when A and B are both whole numbersList the factors of A.List the factors of B.Find the greatest common factor of A and B, GCF(A, B)Divide A and B each by the GCF.Use the whole number results to rewrite the ratio in simplest form.
## What is 12 60 as a percentage?
20%Convert fraction (ratio) 12 / 60 Answer: 20%
## How do you simplify 24 60?
Thus, 2/5 is the simplified fraction for 24/60 by using the GCD or HCF method. Thus, 2/5 is the simplified fraction for 24/60 by using the prime factorization method.
## What is the ratio of 24 to 60?
40%Latest decimal numbers, fractions, rations or proportions converted to percentages24 / 60 = 40%Nov 07 18:50 UTC (GMT)15.5 / 20 = 77.5%Nov 07 18:50 UTC (GMT)547 / 700 = 78.142857142857%Nov 07 18:50 UTC (GMT)All decimal number, fractions, ratios or proportions converted to percentages10 more rows
## What is the simplest form of 25 60?
What is 25/60 Simplified? – 5/12 is the simplified fraction for 25/60.
## What is the simplest form of 16 60?
What is 16/60 Simplified? – 4/15 is the simplified fraction for 16/60.
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See what questions
a doctor would ask.
About 400,000 of the 1.1 million worldwide sufferers of multiple sclerosis live in the US. It is an autoimmune disease which affects the brain and spinal cord. Hard scar tissue replaces the myelin sheaths of neurons which results in disruption of nerve impulses. The condition is progressive and degenerative. Hereditary factors account for about 20% of MS cases. Symptoms of MS can vary and are often misdiagnosed. Common symptoms include fatigue, dizziness, loss of coordination, loss of balance, difficulty walking, bladder dysfunction, bowel dysfunction, sexual dysfunction, double or blurred vision, ophthalmic pain or vision loss, tingling in hands or feet, numbness in hands or feet and weakness in arms or legs. Other less common symptoms include seizures, hearing loss, dysphagia, difficulty swallowing, itching, headache and muscle spasms. Secondary symptoms may include chronic urinary infections, slurred speech, forgetfulness, confusion, weakness, bone density loss, difficulty breathing. Bedsores become common as the disease progresses and the patient is less mobile and depression is prevalent at all stages of MS. Some patients are not affected badly and can lead normal lives. There is no cure but treatment may include corticosteroids, beta interferons or tizanidine hydrochloride. Other medications may be used to treat the various symptoms that occur such as fatigue and bladder problems.
Source: summary of medical news story as reported by Express Newsline
About: Multiple sclerosis occurs in 1.1 million people worldwide
Date: 13 January 2005
Source: Express Newsline
http:/ This summary article refers to the following medical categories:
Related Medical Topics
More News Topics
This summary article refers to the following medical categories:
Search Specialists by State and City
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# Module 2:2 - ANOVA and the General Linear Test
## Fisher-Snedecor Distribution
The Fisher-Snedecor Distribution describes what would happen in nature if you were to take a variance estimate from one sample, then take another sample from the same population and form another variance estimate. This allows us to test the τj's (or taus) from our sample data to the population data. We can see all at once whether our treatment offsets are actually 0.
Fisher-Snedecor
When you take the ratio of two variance estimates, the distribution follows a specific function. The distribution of the ratio of variances is distributed as the F distribution.
Remember
Degrees of Freedom (df): Independent pieces of information that are used in the F distribution
As we have learned before, larger samples produce less skewed sampling distributions of variance. These are all unbiased in that all have a mean that is equal to the population variance. This skew however is very important because it gives the F distribution its characteristic shape.
We can use the F distribution to determine whether all treatment offsets (or τj's) are equal to zero. It will tell all at once whether it's likely to be zero or likely to not be zero. All τj's will be non-zero due to sampling error alone.
Fisher-Snedecor as a ratio of variances
Based on our H0, the variance of the τj's = 0. This is the same as saying every observation is identical. We can test whether we think population τj's are different from zero or not based on sample data.
To do this we must form a test statistic as a ratio of variances.
By this we are able to tell whether treatment offsets (τj's) are reasonably different from what we would expect just by chance error.
### Graphing the F Distribution
The F distribution is a ratio of chi-square distributions. in other words, when we take a sample, estimate a population variance and repeat we get a chi-square distribution. The Fisher-Snedecor distribution is the shape of the ratio.
Distribution of F50,50
This distribution is not symmetric as you are not equally likely to get values above 1 as below 1. This distribution has a skew related the the skew of variance estimates. This skew will depend on how many observations we have and the shape depends on the sampling distributions of variance estimates.
Distribution of F2,50
If the ratio consists of a small number and a large number, there will be many F ratios close to 0 and 1. The average of the distribution (the mean) will still be the same but because the shape of sampling distributions of variance estimates, the shape of the F distribution will change. This change will dramatically depend on how many observations go in to the numerator and denominator estimate.
### Interpreting the F Distribution
If variance of treatments from population is very large and we take a sample from it, than the variance of treatment to variance of error should be larger than 1.
By doing this, we should be able to capture what we expect by chance and what we actually observe.
The numerator is actually capturing two things: systematic variance + random variance if H1 is true (Not all τj's = 0)
F Equation if H1 is true
If there actually is some diff between τj's and population, then the variance in the numerator will tend to be very large.
If there are no true effects, H0 is true and our equation is
random variance/random variance
F Equation if H0 is true
By this equation we know what values of F we should expect to get If we get a value that is very different, we should be able to reject H0 We can also get a p-value to find how unlikely a value is if H0 true
F Equation if H0 is true
We can see our final equation by looking into the SSerror and SStreatment as well as the dferror and dftreatment
SSerror - deviations between actual scores and individual scores squared (how much spread out around group means)
(SSwithin): within a group or amount of error within own group
MStreat :deviation among the τj's divided by degrees of freedom SStreat sum of squared diffs for each individual predicted score from grand mean
The sum for each individual of predicted score(group mean) - grand mean SSbetween - SS between groups vs SSwithin
## Sums of Squares in ANOVA
### Partitioning the Sums of Squares
Every individual in a data set has a value for treatment deviation and error deviation; we are partitioning each individual’s score into one part treatment and one part error.
The calculation for SSerror and SStreatment share a common variable: ŷij, or the predicted score for an individual. Note that this variable is just the individual’s group mean without taking individual error into account. By adding up the equations for calculating SSerror and SStreatment, we get the equation for calculating SStotal. Notice that by canceling out the common variable ŷij, we are left with Yij - Y, which represents how much an individual’s score deviate from the grand mean.
Analysis of variance (ANOVA) helps us understand how big the treatment effect is relative to error.
#### Example
Consider 2 individuals in the cost of flight example: Y(10)(1) and Y(10)(2).
In the first graph, we see the distance between their individual scores (orange dots) and the grand mean (blue dots), which represents each individual’s total deviation. The second graph shows the distance between their individuals scores (orange dots) and their own group means (dark blue dots), which is also known as error deviation. The third graph shows the distance between their own group means (dark blue dots) and the grand mean (blue dots), which is also known as treatment deviation. The total deviation is perfectly cut up into one part error deviation and one part treatment deviation.
### Understanding SStreatment
SStreatment is the same predicted score for every individual in the same group, because it is simply the treatment offset between the group’s mean and grand mean. Therefore, the equation for calculating SStreatment can also be written as:
• nj = number of individuals in group j, and
• tj = group j's treatment offset between group j’s mean and grand mean
## Degrees of Freedom
• Degrees of freedom represents how much independent information we have.
• In ANOVA, we are allocating some of the independent information to SStreatment, and some to SSerror. Similar to sums of squares, the total degrees of freedom we have in a data set (DFtotal) is going to be one part DFerror and one part DFtreatment (see figure).
• Like the sums of squares, the degrees of freedom add up.
• DFerror + DFtreatment = DFtotal
### Degrees of Freedom for SStotal (DFtotal)
• Recall that we use SS/n-1 to calculate variance in a sample. The n-1 here represents the degrees of freedom for SStotal.
• We use n-1 because we have to calculate the grand mean in order to calculate the sums of squares and the variance, and calculating the grand mean takes away one degree of freedom, or one independent piece of information.
• For example, if we have 15 pieces of independent information in a data set, we have 15 degrees of freedom to start with. However, after calculating the grand mean, we only have 14 pieces of independent information, because we can calculate the last piece of information by using the grand mean, so that piece of information is no longer independent. Our degrees of freedom becomes 15-1=14.
### Degrees of freedom for SStreatment (DFtreatment)
• DFtreatment = j-1, where j=the number of groups.
• This is because the treatment offsets for all groups have to add up to 0, because we found these offsets from the grand mean. Therefore, we only have j-1 degrees of freedom or pieces of independent information.
• For example, if there were three groups, and we knew the treatment offset of two groups, we can calculate the third treatment offset because the sum of these offsets is equal to 0. Therefore, the third treatment offset is not independent information, and we have 3-1=2 degrees of freedom.
### Degrees of freedom for SSerror (DFerror)
• DFerror uses up the remaining degrees of freedom we have and can be calculated in two ways:
• The first equation calculates DFerror by subtracting one degree of freedom from the number of individuals in each group (for similar reason as calculating for DFtreatment), and add across all groups.
• The second equation calculates DFerror by subtracting DFtreatment (j-1) and the one DF we used for calculating the grand mean from the total number of individuals in the data set.
## General Linear Test Approach
General Linear Test Formula
In the general linear test, we add parameters, and see how much error we produce by doing so, per parameter added. We compare the amount of error in the full model versus the amount of error in the reduced model. It is a model comparison.
To do so, we...
-Specify a full (unrestricted) model and determine amount of error
-Specify a reduced (restricted) model and determine amount of error
-Test the reduction in error, and divide that by the number of added parameters.
-Divide everything by the baseline error. We have error from the full model, and the restricted model. We use the error from the full model. This is because when the null hypothesis is true, the error from the full model is simply random. So this is the best, unbiased guess of the population variance, since it is uncontaminated by treatment offsets (if the treatment is actually even there).
``` -With this test, we still find an F-statistic, and this test ends up being identical to the analysis of variance test.
```
General Linear Test Formula
Full Model v Reduced Model
Full model
``` Same as one factor linear model
```
Yij = mean + treatment offset + individual error
Reduced model
``` Under the null hypothesis, hold constant the tau sub j’s to be zero
```
yi= grand mean + error
Full Model v Reduced Model
We want to test whether the full model fits our data better than the reduced model.
What we are trying to determine, is for every parameter added, have we reduced error enough to think that it is simply sampling error alone.
Further Explanation of Each Component the Formula
Reduction in error: The reduced model will always have more error, so we take the sums of squares error of the reduced model, and subtract the sums of squares error of the full model (to get a positive value).
Added parameters: The reduced model will always have fewer parameters than the full model, so the difference in degrees of freedom of the two models will give us the number of added parameters. Again, we start with the reduced model, and we take the degrees of freedom from the reduced model, and subtract the degrees of freedom from the full model.
Baseline error: This is the amount of error we should expect in the world. We have two different measures of error, the error from the restricted and the full models. Again, we use the error from the full model. This is because when the null hypothesis is true, the error from the full model is simply random. So this is the best, unbiased guess of the population variance, since it is uncontaminated by treatment offsets (if the treatment is actually even there).
Further Explaining the Components of the General Linear Test Formula
Graphically:
We want to see if the treatment offsets (how they differ from the grand mean) are more different/larger than what random grouping would be.
Graphing Groupings and Showing their Deviations
Assuming we have the same number of observations, the more groupings we have, the more likely we are to see deviations from the grand mean. The whole purpose of the general linear test is to see how much we produced error, per parameter added.
How the General Linear Test Compares to the Analysis of Variance
Further Explaining the Components of the General Linear Test Formula
When we take a look at the numerator, we have the sums of square error from the reduced model, minus the sums of square error from the full model
All of that is divided by the degrees of freedom from the reduced model, minus the degrees of freedom from the full model
The sums of squared and degrees of freedom from the reduced model is really just the sums of squares and degrees of freedom total, and the sums of squared and degrees of freedom from the full model are really the sums of squares and degrees of freedom error. We know that total - error = treatment. This simply turns the entire numerator into the mean square treatment.
Explanation of the Numerator - Step 1
Explanation of the Numerator - Step 2
Explanation of the Numerator - Step 3
With the denominator, we have the baseline error, for which we used the full model to represent. In the denominator, we have the sums of square error from the full model divided by the degrees of freedom from the full model, this is equivalent to the mean square for error we developed in analysis of variance test.
Explanation of the Denominator
Therefore, the entire formula is simplified to mean square treatment over mean square error.
Simplified Formula
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Learning is how the unknown becomes known – how the unexperienced becomes experienced – learning for understanding or ability (as distinct from recall) always involves some degree of stretching attention through confusion (see: Cycle of Engagement). In relation to any particular learning objective, the trick is to minimize the confusion extraneous to the objective and to vivify the confusion within it.
From Children of the Code’s “Changing Trajectories“
The most important point in this piece is the suggestion that we can benefit by “strategically inducing confusion”. As our primary intention in instruction is to inspire and support the learner’s inner (inside-out) participation in learning, leading (pedagogically-strategically conducting) them into confusion means we can meet them in their confusion – be together in their confusion – ‘sync-up’ in their confusion. For both their learning and ours, confusion is a great source of intelligence. For them it represents internal information that something is not right – that something is missing – that they need something more. If, rather than blowing it off and dulling out, they realize the importance of learning into their confusion (rather than avoiding it) their confusion can lead them to clarify what they need to progress (See: ‘meaning needs‘ ). For educators and the educational system, understanding where, how, and why learners are confused is the best possible source of information for designing and/or tuning instruction. (See: Miraculous Intersection).
Beyond the purely cognitive pros and cons of confusion, learners have emotional reactions to the feeling of confusion. In particular, when learners feel ashamed of themselves for being confused their emotional response can disable their ability to learn (through whatever is confusing them). (See: Confused?: shame on you!)
Sidney D’Mello, one of the authors of the study said: “It is also important that the students are productively instead of hopelessly confused”. And, “it is not advisable to intentionally confuse students who are struggling”. If only we could get that one. We may not intend to confuse students but we are oblivious to many ways we unnecessarily confuse them (see: Cycle of Engagement). For example:
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Conservation of Snow Leopards
Europe/Middle-East > North Asia/Mongolia > Mongolia
As human population expands and natural habitats shrink, people and animals are increasingly coming into conflict over living space and food. The impacts are often huge. People lose their crops, livestock, and sometimes their lives. Snow leopards use domestic livestock as a food resource in nearly all areas where they overlap with resultant retribution killing by herders.
Although snow leopards have been coexisting with humans (who probably always have hunted the superb cats) for thousands of years, their numbers are believed to have diminished recently, as a consequence of increasing conflicts with humans. During the last 3 years, attacks on livestock seem to have increased in many parts of the snow leopard’s range. Therefore local people and decision makers think that snow leopard populations have increased, although this is the result of decreasing wild ungulates and increasing competition to livestock.
Snow leopard are legally protected in Mongolia, but the population is believed to decline because of :
1) Poaching for fur, bones and body parts to be sold on the black market.
2) Loss of preys as a result of (illegal) over-hunting of ibex, argali, and marmots, and competition with livestock.
3) Loss of habitat as a consequence of degradation (over-grazing) and fragmentation.
4) Retaliation killing of stock raiders.
5) Lack of awareness and support of local people for the conservation of snow leopards, their preys and habitat (Evans et al. 2003).
WWF Mongolia has started a snow leopard conservation project in 1997 in the Uvs aymag (province), one of the aymags of the Altai-Sayan ecoregion in Western Mongolia. The project has achieved a decline in illegal hunting of snow leopards in its implementation area. Public awareness and education campaigns have been highly effective, but in Gobi-Altai, Khovd and Bayna-Ulgii aymags, snow leopards are killed out of retaliation for ‘stealing’ livestock, and then pelts are illegally traded. 33 snow leopard pelts have been traded from Mongolia to Russia during the last 5 years.
Reduce human-snow leopard conflicts.
- Change attitudes and behaviour of public, and increase level of knowledge on snow leopards.
- Reduce illegal hunting of snow leopards and prey species.
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Summary: Diagnosing autism has never been easy since screening tests rely on analyzing a child’s behavior. But researchers have a developed a new type of screening test that can both diagnose and treat the condition in kids over age three. What makes this new test so revolutionary is that it uses a much more objective measurement than current tests use: A child’s involuntary movements.
By Sharon Mazel | Posted: July 26, 2013
Autism — a developmental disorder that hinders communication and the ability to interact socially — affects approximately one in every 50 children in this country. Traditional tests to screen for the condition focus mostly on analyzing a child’s behavior (such as repetitive behaviors or misunderstanding social cues). The problem with such tests is that they’re based mostly on observation — a therapist asking a child questions and observing his social and communication skills. In other words, it’s very subjective. What’s more, treatment is difficult because it attempts to condition a child to reading external signals in order to behave in a socially acceptable way, as opposed to teaching the child self-motivating skills that will enable him to behave in a socially acceptable manner. That’s why a new screening test and treatment that promises to objectively diagnose the condition and then teach self-motivation is making headlines.
Scientists at Rutgers University and Indiana School of Medicine developed a computerized tool that tracks and analyzes the pattern of a child’s random involuntary movements. Since children with autism don’t follow predictable patterns when it comes to involuntary movements, the experts were able to objectively diagnose the condition by looking at how much the patterns of motion of those with autism differed from those who were developing more typically. The researchers also conducted tests that found that this screening tool can be used as therapy, helping autistic children learn and communicate more effectively.
Here’s how it works: Sensors placed on an autistic child’s body measures 240 measurements per second, capturing data on his involuntary random movements. Since those with autism have very different involuntary movements than those without the condition, the experts were able to determine who has the condition.
The researchers tested nearly 80 children and adults with autism (including those with mild and more severe forms of the condition) and found that the screening technique correctly diagnosed the patients each time. They published their findings in the journal Frontiers in Neuroscience.
They then used the test as an early intervention therapy for children by connecting the children to the sensors and giving them computer games to play. In the first game, they had to match two geometric shapes on the computer screen by pointing at the correct shape. When they performed the computer game task correctly, a fun video would play. In another game, the children had to figure out a position in the air to keep their hand, which would then prompt the video to play — sort of like when you play on the Wii. These games taught the children how to self discover which movements they needed to do to get what they wanted. When the children returned weeks later to play the games, they remembered the simple motion they needed to do in order to get the videos to play on the screen.
What makes this type of treatment work? The ultimate goal of the treatment is to enable autistic children to learn socially acceptable behaviors on their own. If children are encouraged to perform certain tasks through self-motivation, rather than being told what to do, they’ll have an easier time self motivating themselves to learn new things in general, including socially acceptable behaviors.
Some experts predict this new technique will revolutionize the way autism is detected and treated... but it’s likely to take a long time before the screening tool is available to therapists and doctors. Still, the researchers predict that parents of autistic children will be able to easily adopt the computerized methodology to work with their children, regardless of whether the methods become available publicly. If your child is autistic, speak to his therapist and doctor about this novel approach to find out if there’s a way you can adopt some of its techniques to help your child.
Photo credit: Flickr3 more things to read
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How it works
Place between five and ten pieces of large paper with a statement on around the room, depending on how much time you have and the ability of your pupils.
Pupils are allowed to go wherever they wish and write a sentence or paragraph either for or against the statement. The first time you do this it may be useful to outline the language you would encourage your pupils to use and give them sentence starters and phrases to build or refute an argument.
Pupils should visit each one or as many statements as they can in the time available and debate in writing, and in silence, whether tourism is beneficial for, or negatively, impacting the Arctic. Encourage pupils to not just write yes/no, but also to give reasons to support their viewpoint and respond to comments made by their peers.
Once they have worked at two or three different stations, remind pupils to refute the arguments that other pupils have already written.
After the allocated time has lapsed, allow a few extra minutes for pupils to revisit the sheets that they contributed to earlier in the activity to see if the subsequent contributions support or challenge their initial thoughts and whether their initial thinking has been influenced.
Feedback into a whole class discussion.
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What is Fiscal Policy?
Fiscal policy is initiated by the legislative and/or the executive branches of government to change the direction of the economy. It is an economic intervention by the government to change the policies that exist in the economy. This can happen through changes in the level of taxation and government purchases in order to stimulate economic activities. The tools of the government are government expenditures i.e. spending of government on public goods and services and taxes. Fiscal policy aims at a stable economic growth and development for the country, full employment equilibrium, stability in prices and balance of payments equilibrium.
The fiscal policy of a country generally focuses on increasing the growth rate of the economy and also keeping the inflation within limits. Fiscal policy affects the aggregate demand through an increase in expenditure or through taxes. Therefore, any change in fiscal policy gets reflected on the aggregate demand curve. The aggregate demand consists of consumption, investment, government spending and net exports. Any change in the components gets reflected on the position and slope of the curve.
AD = GDP = C(Y - T) + I + G + NX
Here, C= consumption Y = income, T = taxes, Y - T = disposable income, I= investment, G= government spending, NX= net exports
Consumption is a function of disposable income and assumes G and T are exogenous. Fiscal policy affects the growth rate of aggregate demand, given a constant growth in aggregate supply. There are two types of fiscal policy: Expansionary and Contractionary.
Expansionary Fiscal Policy
Expansionary fiscal policy is enacted as a response to recessions or employment shocks through an increase in government spending on infrastructure, education, and unemployment benefits etc. This also stabilizes the employment in the economy and helps the economy to move out of the recession. Here, the budget deficit increases. The aggregate demand curve shifts right.
Also, if there is a recessionary gap in the economy i.e. the actual output is less than the potential output at full employment, then an expansionary fiscal policy can help to stimulate demand and shift the AD curve to the right. And this is mainly done through an increase in government spending or cut taxes.
An increase in government spending directly affects the AD curve and on the other side, a cut in taxes will indirectly affect AD curve because consumers will have more money in their pockets after taxes.
Here, the economy is at point-A with Y1 level of output and P1 level of price. An expansionary fiscal policy shifts the AD curve to the right from AD to AD2 resulting in an output increase from Y1 to Y2 and increase in the price level from P1 to P2.
The main rationale behind such policy is that with more government spending and cut in taxes, people will have more money in their hands which lead to more spending and creates demand resulting in business expansion and the creation of more jobs. It helps in restoring consumers and businesses confidence.
When there is higher government spending with lower corporate taxes, then small firms will enjoy greater sales and pay fewer taxes resulting in higher profits. Also, more government spending means more government jobs resulting in more consumers spending. Thus, overall there is more growth.
Limitations of Expansionary Fiscal Policy
- It puts pressure on the budget deficit of the country and increases debt.
- The effectiveness of the policy depends on the timing and accurate forecasting of the policy decision.
- There is an adjustment lag in case of expansionary fiscal policy.
Contractionary Fiscal Policy
Contractionary fiscal policy is enacted when an economy is a state of out-of-control growth causing inflation and asset bubbles. This type of policy is used to reduce government spending and debt of the country. It is just the opposite of expansionary fiscal policy.
Here, there is a reduction in government spending or increase in taxes through which the growth slows down. It generates a more sustainable economy.
If an economy is growing at a faster rate, then the actual output overpowers potential output at full employment level and this creates an inflationary gap in the system. Thus, a contractionary fiscal policy removes that gap and brings back the economy at full employment level. Through decrease in government expenditures and increase in taxes, the budget deficit of the country improves. The main rationale behind this policy is when the inflation is too strong, the pace of the economy needs to slowdown. Therefore, through an increase in taxes, the government takes back the extra money from the hands of the people i.e. decrease in circulation of money.
However, such a policy leads to a sluggish economy with higher unemployment levels. Nonetheless, this process continues because the government focuses to even out the business cycles and have a stable economic growth.
Contractionary fiscal policy decreases the aggregate demand because the decrease in government spending and an increase in taxes results in lesser money for consumers to consume or invest. Thus, the AD curve shifts left.
Here, initially, the economy was at point A with price P1 and output Y1. There is a decrease in government expenditures and/or increase in taxes which decreases the aggregate demand which shifts the AD curve to the left from AD1 to AD2. Now, the equilibrium is established at point B having a lower level of output Y2 and lower price level P2. Thus, when there is very high inflation, the government takes up a contractionary fiscal policy to decrease the price level and have stable growth.
Contractionary fiscal policy is painful for small firms because there is less demand for goods and services. Both corporations and consumers spend less because there is less money injected into the system.
Limitations of Contractionary Fiscal Policy
- Higher tax rates have a negative impact on workers. They lose the incentives to work harder, thus productivity decreases.
- The presence of time lags for adjustment of the new policy.
- Reduced government spending might have an effect on the public goods and services.
This curve is a graphical representation of the relationship between tax rates and tax revenues. This curve suggests that the revenues will decline after a certain tax rate.
The horizontal axis shows the tax rate and the vertical axis shows the revenue collected from taxes which go to the government. This curve shows the trade-off which government needs to decide between tax rate and tax revenue. The Laffer curve depicts two type of relationship. First, arithmetic i.e. when the tax rate increases, more revenue will be collected but this happens only till the peak tax rate which maximizes the revenue (T*). Second, economic i.e. tax rates increasing after a certain point (T*) would cause people not to work as hard or not at all because of no incentive is left to work thereby reducing tax revenue. Eventually, if tax rates reached 100%, shown to the far right on this curve, all people would choose not to work because everything they earn will go to the government and nothing will be left with them. Thus, 0% tax rate and 100% generates no revenue at all. The economic effects recognize the positive impact of lower tax rates on work, output, and employment because it provides incentives to increase these activities. However, higher tax rates penalize people for engaging in these activities.
Thus, Laffer curve does not say whether a tax cut will raise or lower revenues, nor it says that any and all tax rate reductions would necessarily bring in total revenues but it tells us that tax rate reductions will always result in a smaller loss in revenues. This means that the higher the starting tax rate, the more effect it will have on the supply-side stimulus. It can be concluded that tax rate cuts will generate growth, jobs and income for all which is desirable for the economy.
Therefore, the government needs to know that optimal tax rate (T*) which will maximize the revenue and also people will continue to work hard.
Secondary Effects of Fiscal Policy
Fiscal policy has some effects on the credit market too. Firstly, when the government takes up an expansionary fiscal policy, deficit increases and to finance it, the government borrows funds, thus, debt increases. However, the money supply is constant, as the amount of borrowed funds increases, money left in the economy for circulation decreases, therefore the interest rates increases. Due to this, investment demand decreases because now the interest rate on loans which are used for investment purposes increases. This effect is called crowding-out effect because an expansionary fiscal policy raises the aggregate demand for the economy but at the same time it crowds out or reduces aggregate investment expenditures and consumption expenditures because these are interest sensitive.
Secondly, similar things happen in the case of contractionary fiscal policy but just in the opposite direction. In the case of a restrictive policy, the government spending decreases budget surplus increase which implies borrowings are reduced. Thus, the supply of available funds in the credit market increases causing the interest rate to fall. Here, there is an overall decline in the aggregate demand but at the same time due to the fall in interest rates, the demand for investment expenditures in private sector and consumption expenditures is stimulated. This is called as a crowding-in effect.
Thus, these effects mitigate the effectiveness of the parent policy and counteract to the shift in aggregate demand curves.
Fiscal Policy and Unemployment
According to Keynesian view, expansionary fiscal policy can be effective in reducing unemployment because as the aggregate demand increases, the demand for jobs increases to produce more output. However, classical view is against this. They argue that fiscal policy only reduces cyclical unemployment because in the long-run, economy will return to its full employment level and an expansionary fiscal policy just causes inflation
Fiscal policy cannot solve the unemployment problem because there exists frictional and structural unemployment and fiscal policy may not be able to solve such problems. Here, in such types of unemployment, the main reason is a lack of skills and training which cannot be solved by a country’s fiscal policy.
Click here to learn how government conducts fiscal policy through taxes
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MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB
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Let $R$ be the real line. Consider the following subsets of the plane $\boldsymbol{R} \times \boldsymbol{R}$
$S=\{(x, y): y=x+1$ and $0<x<2\}$
$T=\{(x, y): x-y$ is an int eger $\}$
Which one of the following is true?
1) Neither $S$ nor $T$ is an equivalence relation on $R$
2) Both $S$ and $T$ are equivalence relations on $R$
3) $S$ is an equivalence relation on $R$ but $T$ is not
4) $T$ is an equivalence relation on $R$ but $S$ is not
| 9 views
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(4)
Explanation
$T=\{(x, y): x-y \in Z\}$
as $0 \in Z T$ is a reflexive relation
If $x-y \in Z \Rightarrow y-x \in Z$
$\therefore T$ is symmetrical also If $x-y=Z_{1}$ and $y-z=Z_{2}$
Then $x-z=(x-y)+(y-z)=Z_{1}+Z_{2} \in Z$
$\therefore T$ is also transitive.
Hence $T$ is an equivalence relation Clearly $x \neq x+1 \Rightarrow(x, x) \notin S$
$\therefore S$ is not reflexive
by Diamond (75,914 points)
0 like 0 dislike
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The sinking of the Titanic hasn't just become a pop-cultural icon; it's also one of the best studied disasters of all time. But there are hints that it may not have been a typical event. The ship's captain ordered that women and children be allowed to evacuate first, and his officers apparently enforced it by shooting guns in the area of anyone who disobeyed. In contrast, the Lusitania sank so quickly that there wasn't time to organize an evacuation, and women and children fared really badly.
To get to the bottom of this, two Swedish researchers have compiled data on 18 major maritime disasters, ranging from 1852 to 2011. An analysis of the survival shows that, in contrast to the Titanic, the crew generally does the best, and children the worst. Women started out doing pretty poorly, but their survival have been going up as the years have passed. But before we conclude that chivalry was dead and seeing a revival, we'll caution you that it may be just that more women now learn how to swim.
The authors argue that shipwrecks can actually tell us a fair bit about human behavior, since everyone stuck on a sinking ship has to do a bit of cost-benefit analysis. People will weigh their options—which will generally involve helping others at great risk to themselves—amidst a backdrop of social norms and, at least in case of the Titanic, direct orders from authority figures. "This cost–benefit logic is fundamental in economic models of human behavior," the authors write, suggesting that a shipwreck could provide a real-world test of ideas derived from controlled experiments.
Eight ideas, to be precise. That's how many hypotheses the authors lay out, ranging from "women have a survival advantage in shipwrecks" to "women are more likely to survive on British ships, given the UK's strong sense of gentility." They tested them using a database of ship sinkings that encompasses over 15,000 passengers and crew, and provides information on everything from age and sex to whether the passenger had a first-class ticket.
For the most part, the lessons provided by the Titanic simply don't hold. Excluding the two disasters mentioned above, crew members had a survival rate of over 60 percent, far higher than any other group analyzed. (Although they didn't consistently survive well—in about half the wrecks, there was no statistical difference between crew and passengers). Rather than going down with the ship, captains ended up coming in second, with just under half surviving. The authors offer a number of plausible reasons for crew survival, including better fitness, a thorough knowledge of the ship that's sinking, and better training for how to handle emergencies. In any case, however, they're not clearly or consistently sacrificing themselves to save their passengers.
At the other end of the spectrum, nearly half the children on the Titanic survived, but figures for the rest of the shipwrecks were down near 15 percent. About a quarter of women survived other sinkings, but roughly three times that made it through the Titanic alive. If you exclude the Titanic, female survival was 18 percent, or about half the rate at which males came through alive.
What about social factors? Having the captain order "women and children first" did boost female survival, but only by about 10 percentage points. Most of the other ideas didn't pan out. For example, the speed of sinking, which might give the crew more time to get vulnerable passengers off first, made no difference whatsoever to female survival. Neither did the length of voyage, which might give passengers more time to get to know both the boat and each other. The fraction of passengers that were female didn't seem to make a difference either.
One social factor that did play a role was price of ticket: "there is a class gradient in survival benefitting first class passengers." Another is the being on a British ship, where (except with the Titanic), women actually had lower rates of survival.
But the biggest factor seems to have been time. Since World War I, the gender gap has been shrinking, something the authors ascribe to the changing role of women in society. But it's not clear whether that involves some sort of social factor that plays out in disasters or just a better level of fitness overall.
Although the analysis includes citizens of 30 nations among the passengers and crew involved, all but one of the ships in the study belong to the US or European nations. This is probably because the authors required detailed passenger lists to perform their analysis, but it is a fairly narrow window into human behavior. With the sample it has, the study suggests that there aren't many rules when it comes to surviving a sinking ship but, if you're a woman, it's better to rely on equality between the sexes than chivalry.
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An oratorio is a large musical composition art form for orchestra, vocal soloists and chorus, usually with a narration that unifies the dramatic story. It differs from an opera in that it does not use theatrical scenery, costumes, or acting stylizations. The oratorio, however, closely mirrors the opera in musical style and form, except that choruses are more prominent in oratorios than in operas. It was the use of the choruses which gave composers a unique commentation for the depiction of Biblical stories. One of the most known of the oratorios is the 'Messiah' by George Frideric Handel, a massive work reflecting teachings from the New Testament. The peak periods for the composition of oratorios were the seventh and eighteenth centuries when the Baroque period was experiencing its height in the consummation of the grandeur and splendor in its art forms.
Since the word, 'oratorio', was derived from the Italian word for a location for prayer, most oratorios from the common practice period to the present day have biblical themes or strong spiritual subjects. Handel composed oratorios based on themes from the Old Testament such as 'Saul', 'Joshua', 'Israel in Egypt', and 'Judas Maccabaeus'. Yet, Handel and other composers composed secular oratorios based on themes from Greek and Roman mythology. The oratorio usually unfolds under the direction of a speaker or narrator usually with arias, recitatives, duets, trios, quartets, quintets, and choruses. Whether religious or secular, the theme of an oratorio is meant to be weighty, and can include such topics as the creation of the world, the life of Jesus, or the career of a classical hero or biblical prophet.
The plot of an oratorio is often minimal, and some oratorios are not narratives at all. While operas are usually based on a dramatic narrative, in oratorios the aesthetic purpose of the narrative is more often to provide organization and significance to a large musical work. For example, in Handel's oratorios, he has "the chorus - the people - the center of the drama. Freed from the rapid pace imposed by stage action, each scene and concomitant emotions are expanded to vast dimensions. The chorus touches off the action, and then reflects upon it. As in Greek tragedy it serves both as protagonist and ideal spectator. The characters are drawn larger than life-size. Saul, Joshua, Deborah, Judas Maccabacus, Samson are archetypes of human nature—creatures of destiny, majestic in defeat as in victory."
By the mid-seventeenth century, two types had developed:
- The oratorio volgare (in Italian) - with the following representative examples:
- Giacomo Carissimi's Daniele;
- Marco Marazzoli's S Tomaso;
- similar works written by Francesco Foggia and Luigi de Rossi.
Lasting about 30 to 60 minutes, oratorio volgares were performed in two sections and separated by a sermon; their music resembles that of contemporary operas and chamber cantatas.
- The oratorio latino (in Latin) - first developed at the Oratorio del SS. Crocifisso, was related to the church of San Marcello al Corso in Rome.
The most significant composer of oratorio latino is Giacomo Carissimi, whose Jephte is regarded as the first masterpiece of the genre. Like most other Latin oratorios of the period, it is in one section only.
Oratorios usually contain:
- An overture, for instruments alone.
- Various arias, sung by the vocal soloists.
- The recitative, usually employed to advance the plot.
- Finally, choruses, often monumental and meant to convey a sense of glory. Frequently the instruments for oratorio choruses include timpani and trumpets.
List of notable oratorios
(ordered chronologically by year of premiere)
- Johann Sebastian Bach, the Christmas Oratorio (1734)
- Johann Adolf Hasse, "Serpentes ignei in deserto" (1735, 1736 or 1739)
- George Frideric Handel, Israel in Egypt (1739), notable for being the source of the earliest known recording of classical music, made in June 6, 1888 on a wax cylinder.
- Handel, Messiah (1741). This is by far the most familiar and widely performed of oratorios, at least in English-speaking countries.
- Handel, Samson (1743)
- Handel, Judas Maccabaeus (1747)
- Joseph Haydn, The Creation (1798)
- Haydn, The Seasons (1801)
- Felix Mendelssohn, Elijah (1846)
- Hector Berlioz, L'Enfance du Christ (1854)
- Igor Stravinsky's opera, "Oedipus rex" (1927)
- Artur Kapp, Hiiob (Job) (1929)
Advantages of the Oratorio
The oratorio as a large dramatic narrative composition for orchestra, vocal soloists and the chorus were most loved by those who were acquainted with the teachings from the Old Testament. Even though oratorios were large scale productions, oratorios were different from operas in that they were less expensive to produce with no expensive operatic staging, machinery or costumes. Thus they attracted audiences from all economic phases of life, which constantly reinforced the biblical scenarios and stories for all to enjoy.
- Crowther, Victor. The oratorio in Modena. Oxford: Clarendon Press; NY: Oxford University Press, 1992. ISBN 0-198-16255-3
- Machlis, Joseph. The Enjoyment of Music. New York: W.W. Norton & Co. Inc., 1977. ISBN 0-393-09125-2
- Pahlen, Kurt, Weiner Pfister, Rosemarie Konig, and Thurston J. Dox. The world of the oratorio: Oratorio, Mass, Requiem, Te Deum, Stabat Mater, and large cantatas. Portland, OR: Amadeus Press, 1990. OCLC 20220562
- Smither, Howard E. A history of the oratorio. Chapel Hill: University of North Carolina Press, 1977-2000. ISBN 0-807-81274-9
- History and development of the musical form Retrieved September 27, 2007.
- Thiruvasagam In Oratorio by Maestro Ilayaraaja Retrieved September 27, 2007.
- Oratorio Classic Music Pages. Retrieved September 27, 2007.
- Oratorio Official Oratorio Website, Retrieved September 27, 2007.
- Oratorio Answers.com. Retrieved September 27, 2007.
New World Encyclopedia writers and editors rewrote and completed the Wikipedia article in accordance with New World Encyclopedia standards. This article abides by terms of the Creative Commons CC-by-sa 3.0 License (CC-by-sa), which may be used and disseminated with proper attribution. Credit is due under the terms of this license that can reference both the New World Encyclopedia contributors and the selfless volunteer contributors of the Wikimedia Foundation. To cite this article click here for a list of acceptable citing formats.The history of earlier contributions by wikipedians is accessible to researchers here:
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files/ MHCA -- Blogmeister
MHCA Who will continue to use their blogs?
by MHCA teacher: Rye Alumni
Blog Entries
List 25, 50, all
Math Letter Dearest Father,
This year in math, I've learned about graphing, probability and the order of operations in Module 1.We've learned how to interpret bar and line graphs, this helps when I'm recording data. I learned to use a bar graph when the data is separated into different categories and a line graph when you have one category of data that changes over time.
We also took information from frequency tables, which we also learned how to make. These tables are made with data that was already taken. The frequency table consist of three columns, one for the subject category, one for the for tallies and one for the frequency. After this we learned how to how to make frequency tables and bar and line graphs we moved on to using variables to solve number tricks.
We used algebra to solve these number tricks. This section in Module 1 was more complicated but it became easy after many worksheets. In Module 1 we also learned how to use sequences, word sentences, tables and graphs. We did many worksheets and math point pages on this subject. Modeling these graphs, sequences, tables and word sentences was not a challenge for me.
Next, we learned to find the nth term of a number sequence, this was some serious algebra. This was very interesting to learn because I haven’t studied number sequences before. Next the powers flowered and we started using exponents in the next section of Module 1. Exponents are a short way of writing a multiplication problem. We wrote both standard and exponential forms in this Module.
When we finished exponents we started learning about probability. We did many experiments with flipping coins and picking little different colored squares out of a bag. I also did a project with my friend on probability. We found the experimental and theoretical probability. We picked jelly beans out of a bag, then we wrote about our results in a paragraph and table. I enjoyed this project because it was really active and I got to eat jelly beans! Our experiment was called Jelly Belly.
The last thing we learned in Module 1 is the order of operations. In Module 1, we learned an easy way to remember the order of operations: PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction). The order goes left to right in the PEMDAS order. I learned a lot of useful and handy information in Module 1.
Sincerley,
MHCA
Article posted October 22, 2010 at 11:21 AM • comment • Reads 892 • Return to Blog List
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Diabetes is a major public health concern. The global prevalence of diabetes is approximately 9% among adults, while annually almost 1.5 million deaths are attributed to diabetes and its complications. The WHO projects that diabetes will be the 7th leading cause of death by 2030.
In excess of 90% of diabetic patients have Type 2 diabetes (formerly called non-insulin-dependent or adult-onset) which results from the body’s ineffective use of insulin, a hormone that regulates blood sugar. The primary causes of Type 2 diabetes include unhealthy dietary patterns, excessive body weight and inadequate physical activity. High blood sugar levels, also known as hyperglycemia, is a common consequence of uncontrolled diabetes and over time leads to serious damage to several organs, increasing the risk of heart disease and stroke, as well as other diabetic complications, such as peripheral neuropathy, diabetic retinopathy, and chronic kidney disease.
The onset of Type 2 diabetes can be effectively prevented or delayed with the adoption of simple lifestyle measures, including achieving and maintaining a healthy body weight, consuming a healthy diet, being physically active, and avoiding tobacco use. Additionally, healthy eating is a vital component for effectively managing diabetes and avoiding disease complications.
According to the International Diabetes Foundation, health awareness and access to affordable healthy food is essential for reducing the global burden of diabetes. The Prolepsis Institute contributes to the prevention of Type 2 diabetes by implementing two public health intervention programs, as well as various awareness and health promotion campaigns in schools, organizations, and the private industry.
In particular, the Prolepsis Institute’s Program “Eu Dia…Trofin” is dedicated to raising widespread awareness of the National Dietary Guidelines, which are tailored for specific population groups, including children and adolescents, adults, and women during pregnancy and lactation. The National Dietary Guidelines are freely accessible at http://www.diatrofikoiodigoi.gr.
In addition, the Prolepsis Institute’s “DIATROFI” Program for Food Aid & Promotion of Healthy Nutrition is currently implemented for the fifth consecutive year. The “DIATROFI” Program actively addresses the problem of hunger and food insecurity among youth attending schools in underprivileged areas throughout Greece. The Program includes the daily provision of healthy meals, tailored to participating students’ age-specific nutritional needs. Daily meals are distributed universally to all students attending schools participating in the Program, with over 11 million meals having been cumulatively offered to 75,000 students in approximately 450 schools nationwide. Additionally, awareness and health promotion campaigns aimed at endorsing healthy eating habits and lifestyles among both students and their families are conducted through the dissemination of educational materials, activities and informative events. The Program “DIATROFI” is implemented in collaboration with the University of Athens School of Medicine, as well as several other Universities in Greece and the U.S.A.. The Program is managed under the auspices of the Ministry of Education, Research and Religious Affairs, and is primarily funded by the Stavros Niarchos Foundation.
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# Chapter 4 Review: More About Relationship Between Two Variables
## Presentation on theme: "Chapter 4 Review: More About Relationship Between Two Variables"β Presentation transcript:
Chapter 4 Review: More About Relationship Between Two Variables
Group Members: Qianya Meng Nikta Kheiri Min Kim 1st period 12/14/11
The Big Idea Transform the graph to achieve linearity
Transform exponential graphs: π¦=π π π₯ to achieve linearity and come up with a transformed equation for the use of extrapolation. Transform power functions π¦=π π₯ π to achieve linearity and come up with a transformed equation for the use of extrapolation. Learn to use marginal distribution and conditional Recognize relationships between two variables.
Vocabulary You Need to Know
Transforming or re-expressing the data is applying a function such as the logarithm or square root to a quantitative variable Log Rules: 1) logb(mn) = logb(m) + logb(n) 2) logb(m/n) = logb(m) β logb(n) 3) logb(mn) = n Β· logb(m)
Vocabulary Linear growth increases by a fixed amount in each equal time period. Exponential growth model Log y = log a + (log b)x Predicted y = ab^x Power law model Log y = log a + p log x Predicted y = ax^p
Vocabulary Two-way table describes two categorical variables
Marginal distributions are the total in each column and row variable Conditional distributions of column variable, given row variable Conditional distributions of row variable, given column variable Simpsonβs paradox is a reversal that an association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group
Vocabulary Causation: Changes in x cause changes in y
Common response: Changes in both x and y are caused by changes in a lurking variable z Confounding: The effect (if any) of x on y is confounded with the effect of a lurking variable z
Key Topics Covered in this Chapter
Modeling nonlinear data Relations in categorical data Establishing causation
Formulas You Should Know
Exponential growth model Log y = log a + (log b)x Predicted y = ab^x Power law model Log y = log a + p log x Predicted y = ax^p
Calculator Key Strokes
Exponential growth modeling Enter the explanatory data into L1 and response data into L2 Draw the scatterplot y versus x Define L3 as the (natural) logarithm of L2 then make a scatterplot of (ln) log versus L1 Perform the least-squares regression on the transformed data Draw the scatterplot Plot the residuals versus L1 With the regression equation in Y1, define Y2 = e^(Y1) or Y2 = log^(Y1).
Calculator Key Strokes
Power law modeling Enter the explanatory data into L1 and response data into L2 Draw the scatterplot y versus x Define L3 as the (natural) logarithm of L1 and define L4 as the (natural) logarithm of L2 Plot L4 versus L3 Calculate the regression equation for the transformed data and store it in Y1 Construct a residual plot Define Y2 as (10^a)(x^b) or (e^a)(x^b) Plot Y2 and the scatterplot for the original data together To make a prediction for the value x = k, evaluate Y2(k) on the home screen
Helpful Hints When the explanatory variable is years, transform the data to βyears sinceβ so that the values are smaller and donβt create overflow problems when you perform the inverse transformation If there is a clear explanatory/response relationship, compare the conditional distributions of the response variable for the separate values of the explanatory variable Even when direct causation is present, it is rarely a complete explanation of an association between two variables
Depths (m) Light intensity 5 168.00 6 120.42 7 86.31 8 61.87 9 44.34 10 31.78 11 22.78 Q1 Some college students collected data on the intensity of light at various depths in a lake. Here are their data: Make a scatterplot suitable for predicting light intensity from depth. Describe the form of the relationship. To verify that the decrease in light intensity follows an exponential model, calculate the ratio of light intensity at consecutive depths. Start with /168.00= what do you conclude? Take the natural logarithm(ln) of the light intensity measurements and plot these values against the corresponding depth. Does this transformation achieve linearity? Calculate the least-square regression equation for the transformed data. Interpret the slope and y intercept of this equation in this setting. Construct and interpret a residual plot. Perform the inverse transformation to express light intensity as an exponential function of depth in the lake. Display scatter plot of the original data with the exponential model superimposed. Is your exponential function a satisfactory model for the data? Use your model to predict the light intensity at a depth of 22 meters. The actual light intensity reading at the depth was .58 lumens. Does this surprise you?
Answer Q1 A) the relationship is strong, negative, and curved.
B) the ratios are all 0.717, so an exponential model is appropriate. C) it achieves linearity. D) if x= depth and y=ln(light intensity), then π¦ = x. T5hye i8ntercept, , provides an estimate for the average value of the natural log of the light intensity decreases on average by for each one meter increase in depth. E) the residual plot shows a fairly random scatter and relatively small residuals, so the linear model is appropriate. F) if x=depth and y=light intensity, y=(e^6.789)(e^-.333x). It is a satisfactory model. G) at 22m, the predicted light intensity would be .584 lumens. No, not surprised.
Q2 Some high school physics students dropped a ball and measured its height at various points along its descent. Table 4.3 shows the time since release and the distance the ball had fallen Make a scatterplot suitable for predicting distance fallen from time since release. describe the direction, form, and strength of the relationship. Perform an appropriate transformation to achieve linearity . Then find a least-square regression model for the transformed data. Comment on the quality of your model in (b) by referring to a residual plot and π 2 . Make a scatter plot of the point (time, πππ π‘ππππ ) to see if this transformation works. Then find a least-square regression model for the transformed data. Comment on the quality of your model in (d) by referring to a residual plot and π 2 Use the two models you obtained in (b) and (d) to predict the distance that the object had fallen after 0.47 seconds. Which prediction do you think is closer to the actual value? Why? time distance .16 12.1 .24 29.8 .25 32.7 .3 42.8 44.2 .32 55.8 .36 63.5 65.1 .5 124.6 129.7 .57 150.2 .61 182.2 1189.4 .68 220.4 .72 254.0 261.0 .83 334.6 .88 375.5 .89 399.1
Answer Q2 (a) relationship is curved, strong, and positive.
(b) if x = time and y = distance, predicted y = x^2 (c) r^2 = and the residual plot shows random scatter and fairly small-sized residuals, so this looks like an appropriate model (d) yes. Square-root of the predicted y = x (e) r^2 = and the residual plot show no pattern, which suggest a good model (f) using model from (b): cm. using model from (d): cm
Q3 Here are data from eight schools on smoking among students and among their parents. How many students are described in the two-way table ? What percent of these students smoke? Give the marginal distribution of parentsβ smoking behavior, both in counts and in percents. Calculate three conditional distributions of studentsβ smoking behavior: one for each of the three parental smoking categories. Describe the relationship between the smoking behaviors of students and their parents in a few sentences. Neither parent smoke One parent smoke Both parents smoke Students does not smoke 1168 1823 1380 Student smoke 188 416 400
Answer Q3 A) 5375 students B) 18.7%
C) both parents smoke: 1780, 33.1%. One parent smokes: 2239, 41.7%. Neither parents smoke: 1356, 25.2%. D) student smokes, given both parents smoke: 400/( )= student doesnβt smoke, given both parents smoke: 1380/( )= student smoke, given one parent smokes: 416/( )= student doesnβt smoke, given one parent smokes: 1823/( )= student smokes, given neither parent smokes : 188?( )= student doesnβt smoke, given that neither parent smokes: 1168/( )= students who smoke are most likely to come from families where one or more of their parents smoke.
Q4 Whether a convicted murder gets the death penalty seems to be influenced by the race of the victim. Here are data on 326 cases in which the defendants was convicted of murder Use these data to make a two-way table of defendantβs race vs. death penalty Show that Simpsonβs paradox holds: a higher percent of white defendants are sentenced to death overall, but for the black and white victims a higher percent of black defendants are sentenced to death. Use the data to explain why the paradox hold in language that a judge could understand White defendant Black defendant White victim Black victim Death 19 11 6 Not 132 9 52 97
Answer Q4 A) white defendant: 19 yes, 141 no. Black defendant: 17 yes, 149 no. B) overall death penalty: 11.9% of white defendants, 10.2% of Black defendants. For white victims, 12.6% and 17.5%; for black victims, 0% and 5.8%. C) the death penalty is more likely when the victim was white(14%) rather than lack (5.4%). Because most convicted killers are of the same race as their victims, whites are more often sentenced to death.
Q5 A study showed that woman who work in the production of computer chips have abnormally high numbers of miscarriages. The union claimed that exposure to chemical used in production causes the miscarriage. Another possible explanation is that these workers spend most of their time standing up. Can we conclude that exposure to chemicals causes more miscarriages? Why or why not?
Answer Q5 No. The βnumber of hours standing up at workβ is a confounding variable.
Q6 A study finds that high school students who take the SAT, enroll in an SAT coaching courses, and then take the SAT a second time raise their SAT mathematics scores from a mean of 521 to a mean of 561. what factors other taking the course might explain this improvement?
Answer Q6 The variable βknowledge gained as a result of taking the SAT previously is a confounding variable.
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November is Diabetic Eye Disease Awareness Month, an ideal time to take a deeper dive into the disease we hear so much about. As eye care professionals, one of our areas of concern is that uncontrolled diabetes can affect your vision, and is a leading cause of blindness for individuals under the age of 74.
What is Diabetes?
Diabetes is a metabolic disorder characterized by high levels of sugar, or glucose, in your blood. Having low levels of glucose in your bloodstream is normal, as sugar is the nutrient that provides energy to your body on a cellular level. Your pancreas produces insulin, a hormone that enables cells to convert the glucose in your bloodstream into energy.
However, if your pancreas isn’t making enough insulin or your cells resist the effect of insulin—or if both conditions are present— you’re at high risk for diabetes. Without enough insulin working in your body, you’ll end up with an excess of glucose in your bloodstream, a condition that can cause serious problems.
Diabetes is a chronic, progressive disease. There is no cure, but it can be treated with careful monitoring of your diet, regular testing of blood glucose levels, and medications.
What are the Two Types of Diabetes? There are two different kinds of diabetes:
- Type 1 Diabetes
Sometimes called juvenile-onset or insulin-dependent diabetes, Type 1 diabetes is typically diagnosed before age 30 and impacts 5-10% of diabetics. The primary cause of Type 1 diabetes is low or no insulin production. It’s not clear why some people don’t produce adequate amounts of insulin—it could be genetic or an autoimmune defect. Type 1 diabetics require regular insulin injections throughout their lifetimes.
- Type 2 Diabetes
Known as adult-onset, or non-insulin dependent diabetes, Type 2 diabetes is diagnosed after age 30, typically, and is the more prevalent form of the disease. Between 90-95% of all diabetics are Type 2 diabetics. The cause of this type of diabetes is that the pancreas doesn’t produce enough insulin, or the cells in your body can’t use it efficiently—or both. This results in glucose levels rising uncontrollably in the bloodstream.
What is Pre-Diabetes?
If you’re among the 86 million Americans classified as pre-diabetic, your blood sugar levels are higher than usual but aren’t quite high enough to be diagnosed with diabetes. If you fall into this category, it’s important to take significant steps to prevent the condition from progressing into diabetes. These steps include increasing exercise and activity, losing weight, and altering your diet to focus on foods with a low glycemic index.
Obesity and living a sedentary lifestyle are primary risk factors for diabetes, so lowering body weight and increasing activity level are smart strategies for decreasing your risk.
Some symptoms of diabetes include being overly thirsty, urinating frequently, increased appetite and unexplained weight loss. Having a blood glucose level over 200mg/dL is the diagnostic level for diabetes.
How Does Diabetes Impact the Body?
Too much blood sugar in the bloodstream creates a problem for your body’s blood vessels, causing damage and complications. Diabetes can cause severe damage to your eyes, nerves, kidneys and almost every system in the body. Diabetes also impacts the cardiovascular system, doubling your risk of having a heart attack or stroke.
How Does Diabetes Affect Your Eyes?
Uncontrolled high blood sugar from diabetes can cause vision loss and blindness – either temporarily or permanently.
Temporary blurriness is often experienced when a diabetic patient is changing medications. Having high blood sugar changes the fluid levels in the tissues of your eyes, causing swelling. This can hinder your eyes’ ability to focus, causing blurry vision. When glucose levels are back under control, such temporary blurred vision often goes away.
However, if your body remains in a state of high blood sugar levels consistently, it causes damage to the tiny capillaries and other blood vessels that feed the tissues in the back of the eyes.
Damage to your eyes can begin during pre-diabetes, where your blood sugar is higher than it should be but not yet high enough for a full diabetes diagnosis.
When your optometrist examines the back of your eyes during a dilated eye exam, he or she can see whether your eyes’ blood vessels are functioning normally or have been leaking fluid into the back of the eye and causing swelling. This swelling can lead to high pressure inside the eye, which is dangerous and can cause vision loss.
How do Optometrists Detect Undiagnosed Diabetes?
Optometrists are often the first line of defense in diagnosing pre-diabetes and diabetes, as the sensitive tissues in the eye can show damage early in patients suffering from high blood sugar. These vascular changes in the eye that cause the blood vessels in the back of the eye to leak and damage tissue are a hallmark of diabetes—and can cause blindness.
In 2014, optometrists diagnosed 401,000 cases of diabetes in patients who didn’t know they were suffering from the disease, likely helping to prevent many of these patients from experiencing further health challenges.
This is one of many reasons why it’s important to see your optometrist for a comprehensive eye exam regularly!
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About Cochlear Implant
This information is intended for general information only and should not be considered as medical advice on the part of Health-Tourism.com. Any decision on medical treatments, after-care or recovery should be done solely upon proper consultation and advice of a qualified physician.
What is a Cochlear Implant?
A cochlear implant is an intricate, medical electronic device that is surgically implanted behind the ear to treat hearing loss. A cochlear implant improves hearing by representing sounds better. Patients may not get back normal hearing but they will be able to hear sounds more clearly and will have an improved understanding of speech in environments filled with noise.
How does the Cochlear Implant work?
A cochlear implant consists of internal parts that are implanted beneath the skin and external parts that are worn by the patient. The external part has a speech processor that helps in creating the sensation of sound. The external microphone and processor gathers the sounds in the environment and converts them into electrical impulses.
The cochlear implant sends impulses to the auditory nerve that transmits the signals to the brain. It can bypass the nonfunctional portions of the ear and provide a direct stimulation to the auditory nerve. These signals are then transmitted by the auditory nerve to the brain that perceives the signals as sound.
A cochlear implant does not increase the volume of sound like a hearing aid but it boosts the nervous reaction to sound.
How is the Cochlear Implant surgery performed?
How to prepare for the Cochlear Implant procedure?
- The area behind the ear is shaved and sterilized.
- A two to three inch incision is made. This leads to the opening in the mastoid bone and into the middle ear.
- The receiver-stimulator part of the cochlear implant is place on a depression that is made on the bone.
- The device is kept in place with an enduring suture.
- Through the opening in the mastoid bone, another opening is created in the cochlea to implant the electrodes.
- The electrode is carefully and very slowly placed through this opening. The structure of electrode is designed to arrange the electrodes very close to the ganglion cells to allow the electrical signals.
- When the device is fitted in place, it is tested and the incision is closed with absorbable sutures.
The patient may need to go through certain medical tests such as:
- Thorough examination of the ear
- Hearing aid evaluation
- Physical examination
- General anesthesia preparation
- Psychological test to determine if the patient can deal with the implant
Duration of procedure/surgery : Approximately 2 hours.
It may take longer for younger children due to the small size of their middle ear structures.
Days admitted : For adult and adolescent patients, the cochlear implant surgery is conducted as an outpatient procedure.
Children, however, may require a one-night-stay at the hospital.
Anesthesia : General anesthesia
Recovery : - A bandage around the head needs to be worn during sleep for some time after the surgery.
- The external parts are fitted about one month after surgery when the surgery site has healed. The device is then turned on and mapped, which involves adjusting the speech processor and fixing the stimulation level for each electrode.
- The patient needs to be trained in interpreting sounds that are heard through the cochlear implant for many day or even years.
- Some patients may experience sound sensations that are mechanical or synthetic for a few weeks.
Risks : - Damage to the nerves that causes facial paralysis and taste disturbance
- Dizziness and balance related problems
- Leakage of cerebrospinal fluid
- Loss of residual hearing
- Failure of the device to work
After care : - Patients who require an MRI may need to remove the magnet in the cochlear implant.
- Care should be taken not to get the external cochlear implants wet. They should be removed before showering, bathing or swimming.
- If the device fails, surgery is required to solve the problem.
Learn more about Cochlear Implant
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Every culture has a tradition of oral storytelling. The 35,000-year-old paintings on the walls of the Lascaux Caves are our earliest recorded evidence of storytelling1, and Aesop, a 6th century BC greek slave, wrote tales which even today are used to teach moral behavior to children. Stories are a means to pass on information, values, and knowledge. They provide the structure and framework through which humans sort, understand, relate and file information.2 In short, through stories people learn about the world and themselves.
Throughout time, narrative has been the most natural and fundamental teaching method and it seems that any lesson begun with the phrase “once upon a time” rivets the attention and interest of students. Simply put, stories are how we learn. The progenitors of the world’s religions understood this, handing down our great myths and legends from generation to generation3. Much research is available today to validate the powerful effect storytelling has as a teaching tool and an instrument to enhance motivation, communication and interpersonal skills.
When writing his book Story Proof: The Science Behind The Startling Power of the Story, Kendall Haven reviewed over 350 research studies and, perhaps unsurprisingly, each study agrees that stories are an effective and efficient vehicle for teaching and motivating, and for the general communication of factual information, concepts and tacit information.4 Specifically, it has been shown that material not learned within the context of a story is less likely to be retained,5, 6 whereas stories “engage us. … and help us to understand by making the abstract concrete and accessible”7. The benefits of the storytelling approach to education have been found to apply in very diverse subject areas. These include teaching literacy8, 9 mathematics,10 science11 and history to children,12 and educating professionals in such field as business13, nursing14 and adult education of foreign languages15 to name just a few.
Massachusetts based historian and folklorist, Merrill Kohlhofer uses storytelling to teach history to elementary children, both in schools across New England and at historic sites including the House of Seven Gables and the Peabody Essex Museums. According to Kohlhofer, “Stories can help make what might otherwise seem dry facts and boring, irrelevant events come alive for the listeners. Because the events and characters of stories help create an emotional connection with the listener, the ideas the story carries make a greater impact, and seem both more relevant and more easily remembered and understood. Listening to stories, participating in them, helps develop children’s linguistic skills – well-crafted stories both entice and challenge the listener to love language and its communicative power and serve to model verbal art.”
“I began by asking my listeners [3rd-5th graders] how many liked history – the response was pretty lukewarm. After the question and answer session with which I conclude these programs, I asked the same question – and the response was overwhelmingly enthusiastic. …stories appeal to the child’s verbal intelligence, not something that happens that often these days where we appear to be shifting to a more visual culture.
Stephanie Wilkins, a longtime third and fourth grade teacher at Odyssey Day School in Wakefield, MA, relies heavily on story telling in her classroom. Stephanie describes the power of story as a teaching tool stating, “Sitting and listening doesn’t do it [educate]. If they are just presented with material, it goes in one ear and out the other. Role play and drama, with them making up their own skits and acting out the stories, helps the students learn to handle and utilize concepts. When kids get up and play a part they are going to learn and be more likely to remember.”
Odyssey Day School builds its entire curriculum on the concept of overriding themes and stories. For example, the school -wide theme last year was Milestones: The path from yesterday to tomorrow. When Stephanie’s class was studying the ancient Greeks, instead of just talking or reading about them they became part of the story. Each child researched and played a role of one of the Greek gods or goddesses. The theme was worked into all aspects of the curriculum. In science, they studied astronomy. In math, they learned about the algorithm and how the Greeks used stars to tell time, while in Art they were making sculptures and dioramas of ancient Greek Columns.
Stephanie expounds on the fact that storytelling not only enhances academic knowledge, but “fosters interrelationships between the kids. When they don’t even realize it, they are learning to step out of their own comfort zones and recognize similarities and differences in others, learning from their ideas. They learn to compliment, cooperate, communicate, plan, organize and they learn to listen. The story is not just about me presenting the material, it is a spring board for discussion for asking questions for probing further. It brings it [the teaching] full circle.”
Another place where storytelling is still growing strong and aiding the development of self-esteem, creativity, and team cooporation is at Guard Up Family Swordmanship in Burlington, MA. Guard Up runs summer camps, after-school and weekend programing based on interactive story telling and role-playing with an emphasis on teaching the values of good sportsmanship, teamwork, compassion, honor and courage. Guard Up really brings the story to life through role-playing which is a means of merging the power of stories with the benefits of active learning17. Children of all ages are fully immersed in medieval fantasy stories designed to entertain and educate. The story lines change and adapt based on the behavior of and choices made by the kids. The broad story arcs are planned in advance by a team of counselors, and evolve daily. Campers, as a group, devise strategies, find solutions, and choose their course of action whether defending their city from an invasion of living puppets, or negotiating a peace agreement with a horde of scurvy pirates.
We interviewed four of the Guard Up counselors, Chris, Lauren, Hannah and Joseph, to find out what inspired them, how they utilize the stories as a tool to impart knowledge and some of the surprising paths the stories took based on the actions of the campers, or Heroes, as they are called. They recognize that storytelling is a co-creative process. Although there is a general story arc the counselors know the importance of letting the plot flow in the direction that the kids take. As Joseph explains, “We can’t plan the specific details because it depends on the decisions of the kids. We change the plot based on what the characters are doing.” Lauren agrees “You want to take it where they take it. You don’t want to be so stuck to the plot. You want them to figure it out and feel excited.” Guard up gives the kids the opportunity to design their own reality or as Joseph putt “the kids get to live their dreams”. They design their characters and have a chance to be who they want to be and try out new things. Many of them choose positive attributes and get rewarded for playing them. On the other hand, if a camper decides to, say, fight her own team mates, she learns consequences within the game which makes her not want to do it in the future.
The motto of Guard Up is “courage, honor and compassion. “ Chris, another Guard Up instructor describes how the heros are given many opportunities to choose to display these attributes, such as the option to help other people without getting anything for themselves. Once, for example, when a village was attacked by monsters, the campers stayed by the side of a shopkeeper, protecting her and even giving her their own healing potions when she was injured. When recollecting this tale, Hannah reflects that “these are the real teaching moments”.
Whether participating in adventure at the summer camp, after-school programs, or weekly classes and activities, the children are, as Lauren says, “learning without even knowing they are doing it”. Some knowledge is applicable in the academic sense, for example they learned basic anatomy during a quest to reassemble the body of their village’s mayor – including his nervous system – or utilized mathematics and deductive logic to answer riddles, figure out clues and solve puzzles. Additionally, history is incorporated both through mythology and true historical figures and settings.
Beyond gaining academic knowledge, they are also learning about themselves, social interaction, values and morals. Getting to be the hero they always wanted to be helps them gain confidence. The emphasis on honor, courage and compassion flows through all of the activities. For instance, when they were on the quest to “re-assemble the mayor” they needed to prove they were true of heart before they could retrieve the heart. As Hannah so aptly put it, “We teach kids social skills by letting them explore outlandish possibilities. They find the boundaries of their personality in a safe environment”. They learn how to work together, negotiate, treat others with compassion, and attempt to solve issues through analytic skills instead of aggression. It also gives them a chance to express their emotions, creativity and imagination.
When we first interviewed the campers, many stated that the characters they designed were more creative than they, themselves were. When shown the paradox that they had designed their characters and that all of the character’s actions were coming from their own minds, one camper, Connor, stated enthusiastically, “If you come here I bet you’ll find out that you’re more creative than you think and that you have more talent than you notice.”
When asked, Connor and his fellow campers, Travis, Casey and Ethan offered many different lessons learned, including:
“Sometimes, you can have the best adventures where you don’t do war – do politeness first”
“Honor the game, be truthful, help others, and always try manners before violence. ”
“Teamwork and thinking about problem-solving can help in the real world.”
“The choices you make can really effect what goes on around you.”
Storytelling, besides being perhaps the oldest method of teaching, still plays a vital role in child development. When schools are becoming focused on teaching to standardized tests, it is more important than ever that children still have a way of learning through imagination and participation. If parents are willing to look, there are still great opportunities for children to benefit from this timeless teaching method. Have you spun a story for your kids today?
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Categories
# Arrangement of Resistors and Ohm’s Law
Resistors can be arranged in series and in parallel.
have different potential differences.
When they are connected in parallel, they have the same potential difference but different current.
# Ohm’s Law
The electric current passing through a metallic conductor is directly proportional to the potential difference applied between its end provided temperature and other physical property of the conductor remains constant.
Current =Potential differenceResistanceV=IR
R is a constant of proportionality and depends on the nature of the material. The unit of resistance is ohm.
Calculations
1. A potential difference of 240V is applied to a lamp of 60 ohms resistance. What amount of current will flow in the circuit?
Solution
Current =Potential differenceResistance=24060=4A
1. (a) Calculate the effective resistance in the diagram shown below. (b) Calculate the current flowing in the circuit aboveSolution(a) Solve the parallel first,1RP=1R1+1R21RP=12+121RP=22=1RP=1ΩR=RP+4=1+4=5Ω(b) Total resistance = 5Ω.From Ohm’s law,Current =Potential differenceResistance=205=4A
1. A current of 3A flows in a circuit when a p.d of 24V is applied to it. The resistance across the circuit is
SolutionI=3AV=24VR=?V=IR24=3×RR=243=8Ω
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humpback whales and their songs
How much information the whales exchange via their calls is still unknown, but there have been some intriguing suggestions. In 1999, John Buck and Ryuji Suzuki of the University of Massachusetts, Dartmouth, claimed they had found evidence for a hierarchical grammar like that of human language in the humpbacks' songs; other researchers, however, remain skeptical.
Species descriptionHumpback whales are well known for their long pectoral fins, which can be up to 15 feet (4.6 m) in length. Their scientific name means "big-winged New Englander" as the New England population was the one best known to Europeans. These long fins give them increased maneuverability; they can be used to slow down or even go backwards.
Similar to all baleen whales, adult females are larger than adult males, reaching lengths of up to 60 feet (18 m). Their body coloration is primarily dark gray, but individuals have a variable amount of white on their pectoral fins and belly. This variation is so distinctive that the pigmentation pattern on the undersides of their flukes is used to identify individual whales, similar to a humans fingerprint.
Humpback whales are the favorite of whale watchers, as they frequently perform aerial displays, such as breaching (jumping out of the water), or slap the surface with their pectoral fins, tails, or heads.
In the summer, humpbacks are found in high latitude feeding grounds such as the Gulf of Maine in the Atlantic and Gulf of Alaska in the Pacific. In the winter, they migrate to calving grounds in subtropical or tropical waters such as the Dominican Republic in the Atlantic and the Hawaiian Islands in the Pacific. The Arabian Sea humpback, however, does not migrate, remaining in tropical waters all year.
Humpback whales travel great distances during their seasonal migration, the farthest migration of any mammal. The longest recorded migration was 5,160 miles (8,300 km). This trek from Costa Rica to Antarctica was completed by seven animals, including a calf. One of the more closely studied routes is between Alaska and Hawaii, where humpbacks have been observed making the 3,000 mile (4,830 km) trip in as few as 36 days.
During the summer months, humpbacks spend the majority of their time feeding and building up fat stores (blubber) that they will live off of during the winter. Humpbacks filter feed on tiny crustaceans (mostly krill), plankton, and small fish and can consume up to 3,000 pounds (1360 kg) of food per day. Several hunting methods involve using air bubbles to herd, corral, or disorient fish. One highly complex variant, called "bubble netting," is unique to humpbacks. This technique is often performed in groups with defined roles for distracting, scaring, and herding before whales lunge at prey corralled near the surface.
In their wintering grounds, humpback whales congregate and engage in mating activities. Humpbacks are generally "polygynous" with males exhibiting competitive behavior on wintering grounds. Aggressive and antagonistic behaviors include chasing, vocal and bubble displays, horizontal tail thrashing, and rear body thrashing. Males within these groups also make physical contact; striking or surfacing on top of one another. These bouts can cause injuries ranging from bloody scrapes to, in one recorded instance, death.
Gestation lasts for about 11 months. Newborns are 13 to 16 ft (4 to 5 m) long and grow quickly from the highly nutritious milk of their mothers. Weaning occurs between 6 and 10 months after birth. Mothers are protective and affectionate towards their calves, swimming close and frequently touching them with their flippers. Males do not provide parental support for calves. Breeding usually occurs once every two years, but sometimes occurs twice in three years.
Distribution and habitat
During migration, humpbacks stay near the surface of the ocean. While feeding and calving, they prefer shallow waters. During calving, humpbacks are usually found in the warmest waters available at that latitude. Calving grounds are commonly near offshore reef systems, islands, or continental shores.
Humpback feeding grounds are in cold, productive coastal waters.
Population and conservation statusOnce heavily exploited, the humpback has been protected since the mid-1960s and is increasing in many parts of the world. There are probably now more than 30,000 in the Southern Hemisphere, 15,000 in the North Atlantic and 18,000 in the North Pacific. Four whales per year can be taken by aboriginal subsistence hunters in St Vincent and the Grenadines.
Related categories EXTRATERRESTRIAL AND NON-HUMAN INTELLIGENCE
Source: National Oceanic and Atmospheric Administration
Home • About • Copyright © The Worlds of David Darling • Encyclopedia of Alternative Energy • Contact
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Medically reviewed on September 29, 2016
Male hypogonadism is a condition in which the body doesn't produce enough testosterone — the hormone that plays a key role in masculine growth and development during puberty — or has an impaired ability to produce sperm or both.
You may be born with male hypogonadism, or it can develop later in life, often from injury or infection. The effects — and what you can do about them — depend on the cause and at what point in your life male hypogonadism occurs. Some types of male hypogonadism can be treated with testosterone replacement therapy.
Hypogonadism can begin during fetal development, before puberty or during adulthood. Signs and symptoms depend on when the condition develops.
If the body doesn't produce enough testosterone during fetal development, the result may be impaired growth of the external sex organs. Depending on when hypogonadism develops and how much testosterone is present, a child who is genetically male may be born with:
- Female genitals
- Ambiguous genitals — genitals that are neither clearly male nor clearly female
- Underdeveloped male genitals
Male hypogonadism may delay puberty or cause incomplete or lack of normal development. It can cause:
- Decreased development of muscle mass
- Lack of deepening of the voice
- Impaired growth of body hair
- Impaired growth of the penis and testicles
- Excessive growth of the arms and legs in relation to the trunk of the body
- Development of breast tissue (gynecomastia)
In adult males, hypogonadism may alter certain masculine physical characteristics and impair normal reproductive function. Signs and symptoms may include:
- Erectile dysfunction
- Decrease in beard and body hair growth
- Decrease in muscle mass
- Development of breast tissue (gynecomastia)
- Loss of bone mass (osteoporosis)
Hypogonadism can also cause mental and emotional changes. As testosterone decreases, some men may experience symptoms similar to those of menopause in women. These may include:
- Decreased sex drive
- Difficulty concentrating
- Hot flashes
When to see a doctor
See a doctor if you have any symptoms of male hypogonadism. Establishing the cause of hypogonadism is an important first step to getting appropriate treatment.
The male reproductive system makes, stores and moves sperm. Testicles produce sperm. Fluid from the seminal vesicles and prostate gland combine with sperm to make semen. The penis ejaculates semen during sexual intercourse.
Male hypogonadism means the testicles don't produce enough of the male sex hormone testosterone. There are two basic types of hypogonadism:
- Primary. This type of hypogonadism — also known as primary testicular failure — originates from a problem in the testicles.
- Secondary. This type of hypogonadism indicates a problem in the hypothalamus or the pituitary gland — parts of the brain that signal the testicles to produce testosterone. The hypothalamus produces gonadotropin-releasing hormone, which signals the pituitary gland to make follicle-stimulating hormone (FSH) and luteinizing hormone (LH). Luteinizing hormone then signals the testes to produce testosterone.
Either type of hypogonadism may be caused by an inherited (congenital) trait or something that happens later in life (acquired), such as an injury or an infection. At times, primary and secondary hypogonadism can occur together.
Common causes of primary hypogonadism include:
- Klinefelter syndrome. This condition results from a congenital abnormality of the sex chromosomes, X and Y. A male normally has one X and one Y chromosome. In Klinefelter syndrome, two or more X chromosomes are present in addition to one Y chromosome. The Y chromosome contains the genetic material that determines the sex of a child and related development. The extra X chromosome that occurs in Klinefelter syndrome causes abnormal development of the testicles, which in turn results in underproduction of testosterone.
- Undescended testicles. Before birth, the testicles develop inside the abdomen and normally move down into their permanent place in the scrotum. Sometimes one or both of the testicles may not be descended at birth. This condition often corrects itself within the first few years of life without treatment. If not corrected in early childhood, it may lead to malfunction of the testicles and reduced production of testosterone.
- Mumps orchitis. If a mumps infection involving the testicles in addition to the salivary glands (mumps orchitis) occurs during adolescence or adulthood, long-term testicular damage may occur. This may affect normal testicular function and testosterone production.
- Hemochromatosis. Too much iron in the blood can cause testicular failure or pituitary gland dysfunction, affecting testosterone production.
- Injury to the testicles. Because they're situated outside the abdomen, the testicles are prone to injury. Damage to normally developed testicles can cause hypogonadism. Damage to one testicle may not impair total testosterone production.
- Cancer treatment. Chemotherapy or radiation therapy for the treatment of cancer can interfere with testosterone and sperm production. The effects of both treatments often are temporary, but permanent infertility may occur. Although many men regain their fertility within a few months after treatment ends, preserving sperm before starting cancer therapy is an option that many men consider.
In secondary hypogonadism, the testicles are normal but function improperly due to a problem with the pituitary or hypothalamus. A number of conditions can cause secondary hypogonadism, including:
- Kallmann syndrome. Abnormal development of the hypothalamus — the area of the brain that controls the secretion of pituitary hormones — can cause hypogonadism. This abnormality is also associated with impaired development of the ability to smell (anosmia) and red-green color blindness.
- Pituitary disorders. An abnormality in the pituitary gland can impair the release of hormones from the pituitary gland to the testicles, affecting normal testosterone production. A pituitary tumor or other type of brain tumor located near the pituitary gland may cause testosterone or other hormone deficiencies. Also, the treatment for a brain tumor, such as surgery or radiation therapy, may impair pituitary function and cause hypogonadism.
- Inflammatory disease. Certain inflammatory diseases, such as sarcoidosis, histiocytosis and tuberculosis, involve the hypothalamus and pituitary gland and can affect testosterone production, causing hypogonadism.
- HIV/AIDS. HIV/AIDS can cause low levels of testosterone by affecting the hypothalamus, the pituitary and the testes.
- Medications. The use of certain drugs, such as opiate pain medications and some hormones, can affect testosterone production.
- Obesity. Being significantly overweight at any age may be linked to hypogonadism.
- Normal aging. Older men generally have lower testosterone levels than younger men do. As men age, there's a slow and continuous decrease in testosterone production.
- Concurrent illness. The reproductive system can temporarily shut down due to the physical stress of an illness or surgery, as well as during significant emotional stress. This is a result of diminished signals from the hypothalamus and usually resolves with successful treatment of the underlying condition.
The rate at which testosterone declines varies greatly among men. As many as 30 percent of men older than 75 have a testosterone level that's below the normal range of testosterone in young men. Whether treatment is necessary remains a matter of debate.
The pituitary gland and the hypothalamus are situated within the brain and control hormone production.
Risk factors for hypogonadism include:
- Kallmann syndrome
- Undescended testicles as an infant
- Mumps infection affecting your testicles
- Injury to your testicles
- Testicular or pituitary tumors
- Klinefelter syndrome
- Previous chemotherapy or radiation therapy
- Untreated sleep apnea
Hypogonadism can be inherited. If any of these risk factors are in your family health history, tell your doctor.
The complications of untreated hypogonadism differ depending on what age it first develops — during fetal development, puberty or adulthood.
A baby may be born with:
- Ambiguous genitalia
- Abnormal genitalia
Pubertal development can be delayed or incomplete, resulting in:
- Diminished or lack of beard and body hair
- Impaired penis and testicle growth
- Unproportional growth, usually increased length of arms and legs compared with the trunk
- Enlarged male breasts (gynecomastia)
Complications may include:
- Erectile dysfunction
- Decreased sex drive
- Muscle loss or weakness
- Enlarged male breasts (gynecomastia)
- Decreased beard and body hair growth
Your doctor will conduct a physical exam during which he or she will note whether your sexual development, such as your pubic hair, muscle mass and size of your testes, is consistent with your age. Your doctor may test your blood level of testosterone if you have any of the signs or symptoms of hypogonadism.
Early detection in boys can help prevent problems from delayed puberty. Early diagnosis and treatment in men offer better protection against osteoporosis and other related conditions.
Doctors base a diagnosis of hypogonadism on symptoms and results of blood tests that measure testosterone levels. Because testosterone levels vary and are generally highest in the morning, blood testing is usually done early in the day, before 10 a.m.
If tests confirm you have low testosterone, further testing can determine if a testicular disorder or a pituitary abnormality is the cause. Based on specific signs and symptoms, additional studies can pinpoint the cause. These studies may include:
- Hormone testing
- Semen analysis
- Pituitary imaging
- Genetic studies
- Testicular biopsy
Testosterone testing also plays an important role in managing hypogonadism. This helps your doctor determine the right dosage of medication, both initially and over time.
Treatment for adults
Treatment for male hypogonadism depends on the cause and whether you're concerned about fertility.
Hormone replacement. For hypogonadism caused by testicular failure, doctors use male hormone replacement therapy (testosterone replacement therapy, or TRT). TRT can restore muscle strength and prevent bone loss. In addition, men receiving TRT may experience an increase in energy, sex drive, erectile function and sense of well-being.
If a pituitary problem is the cause, pituitary hormones may stimulate sperm production and restore fertility. Testosterone replacement therapy can be used if fertility isn't an issue. A pituitary tumor may require surgical removal, medication, radiation or the replacement of other hormones.
- Assisted reproduction. Although there's often no effective treatment to restore fertility in a man with primary hypogonadism, assisted reproductive technology may be helpful. This technology covers a variety of techniques designed to help couples who have been unsuccessful in achieving conception.
Treatment for boys
In boys, testosterone replacement therapy (TRT) can stimulate puberty and the development of secondary sex characteristics, such as increased muscle mass, beard and pubic hair growth, and growth of the penis. Pituitary hormones may be used to stimulate testicle growth. An initial low dose of testosterone with gradual increases may help to avoid adverse effects and more closely mimic the slow increase in testosterone that occurs during puberty.
Types of testosterone replacement therapy
Several testosterone delivery methods exist. Choosing a specific therapy depends on your preference of a particular delivery system, the side effects and the cost. Methods include:
Injection. Testosterone injections (testosterone cypionate, testosterone enanthate) are safe and effective. Injections are given in a muscle. Your symptoms might fluctuate between doses depending on the frequency of injections.
You or a family member can learn to give TRT injections at home. If you're uncomfortable giving yourself injections, a nurse or doctor can give the injections.
Testosterone undecanoate (Aveed), an injection recently approved by the Food and Drug Administration, is injected less frequently but must be administered by a health care provider and can have serious side effects.
- Patch. A patch containing testosterone (Androderm) is applied each night to your back, abdomen, upper arm or thigh. The site of the application is rotated to maintain seven-day intervals between applications to the same site, to lessen skin reactions.
Gel. There are several gel preparations available with different ways of applying them. Depending on the brand, you either rub testosterone gel into your skin on your upper arm or shoulder (AndroGel, Testim, Vogelxo), apply with an applicator under each armpit (Axiron) or pump on your front and inner thigh (Fortesta).
As the gel dries, your body absorbs testosterone through your skin. Gel application of testosterone replacement therapy appears to cause fewer skin reactions than patches do. Don't shower or bathe for several hours after a gel application, to be sure it gets absorbed.
A potential side effect of the gel is the possibility of transferring the medication to another person. Avoid skin-to-skin contact until the gel is completely dry or cover the area after an application.
- Gum and cheek (buccal cavity). A small putty-like substance, gum and cheek testosterone replacement (Striant) delivers testosterone through the natural depression above your top teeth where your gum meets your upper lip (buccal cavity). This product quickly sticks to your gumline and allows testosterone to be absorbed into your bloodstream.
- Nasal. Testosterone can be pumped into the nostrils as a gel. This option reduces the risk that medication will be transferred to another person through skin contact. Nasal-delivered testosterone must be applied twice in each nostril, three times daily, which may be more inconvenient than other delivery methods.
- Implantable pellets. Testosterone-containing pellets (Testopel) are surgically implanted under the skin every three to six months.
Oral testosterone isn't recommended for long-term hormone replacement because it might cause liver problems.
Testosterone therapy carries various risks, including contributing to sleep apnea, stimulating noncancerous growth of the prostate, enlarging breasts, limiting sperm production, stimulating growth of existing prostate cancer and blood clots forming in the veins. Recent research also suggests testosterone therapy might increase your risk of a heart attack.
Coping and support
- Prevent osteoporosis. If hypogonadism occurs during adulthood, make lifestyle and dietary changes to prevent osteoporosis. Regular exercise and adequate amounts of calcium and vitamin D to maintain bone strength are important to reduce the risk of osteoporosis. The Institute of Medicine recommends 1,000 milligrams (mg) of calcium and 600 international units (IUs) of vitamin D a day for men ages 19 to 70. That recommendation increases to 1,200 mg of calcium and 800 IUs of vitamin D a day for men age 71 and older. Talk to your doctor about dietary guidelines that are appropriate for you.
- Learn about erectile dysfunction or infertility. The conditions caused by hypogonadism can cause psychological and relationship problems. Know what to expect from these conditions and what to do if new or uncomfortable feelings develop between you and your partner.
Reduce stress. Talk with your doctor about how you can reduce the anxiety and stress that often accompany these conditions. Many men benefit from psychological or family counseling.
Support groups can help people with hypogonadism and related conditions cope with similar situations and challenges. Helping your family understand the diagnosis of hypogonadism also is important.
- Allow time to adjust. Adolescents with hypogonadism may feel as if they don't fit in. Testosterone replacement therapy will trigger puberty. When given at a slow pace that allows time for adjustment to physical changes and new feelings, the therapy decreases the chance of social or emotional problems.
Preparing for an appointment
Although you're likely to start by seeing your family doctor or general practitioner, you may need to consult a doctor who specializes in the hormone-producing glands (endocrinologist). If your primary care doctor suspects you have male hypogonadism, he or she may refer you to an endocrinologist. Or, you can ask for a referral.
The following information will help you prepare for your appointment, and understand what to expect from your doctor.
What you can do
- Note down any symptoms you're experiencing, even if they seem unrelated to the reason you have scheduled the appointment.
- Write down key personal information, including any major stresses, recent life changes, and history of childhood illnesses or surgeries.
- Make a list of all medications, as well as any vitamins or supplements, that you're taking.
- Write down a list of questions to ask your doctor.
Preparing a list of questions for your doctor will help you make the most of your time together. For male hypogonadism, some basic questions to ask your doctor include:
- What's the most likely cause of my symptoms?
- Are there other possible causes for my symptoms?
- What kinds of tests do I need? Do these tests require any special preparation?
- Is my condition likely temporary or chronic?
- What treatments are available?
- What are the side effects of each treatment?
- What treatment do you feel would be best for me?
- What are the alternatives to the approach that you're suggesting?
- I have these other health conditions. How can I best manage them together?
- Are there any restrictions that I need to follow?
- Are there any brochures or other printed material that I can take home with me? What websites do you recommend?
Don't hesitate to ask other questions you have.
What to expect from your doctor
Examples of questions your doctor may ask, include:
- When did you begin experiencing symptoms?
- Have your symptoms been continuous or occasional?
- How severe are your symptoms?
- What, if anything, seems to improve your symptoms?
- What, if anything, appears to worsen your symptoms?
- When did you begin puberty? Did it seem to be earlier or later than your peers?
- Did you have any growth problems as a child or adolescent?
- Have you experienced any testicular trauma?
- What about head trauma?
- Did you have the mumps as a child or teen? Do you recall if you felt pain in your testicles while you had the mumps?
- Did you have undescended testicles as a baby?
- Did you have surgery for a groin hernia or genital surgery as a child?
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CRISPR is causing a major upheaval in biomedical research. Unlike other gene-editing methods, it is cheap, quick and easy to use, and it has swept through labs around the world as a result. Researchers hope to use it to adjust human genes to eliminate diseases, create hardier plants, wipe out pathogens and much more besides. “I've seen two huge developments since I've been in science: CRISPR and PCR,” says John Schimenti, a geneticist at Cornell University in Ithaca, New York. Like PCR, the gene-amplification method that revolutionized genetic engineering after its invention in 1985, “CRISPR is impacting the life sciences in so many ways,” he says....
Biologists have long been able to edit genomes with molecular tools. About ten years ago, they became excited by enzymes called zinc finger nucleases that promised to do this accurately and efficiently. But zinc fingers, which cost US$5,000 or more to order, were not widely adopted because they are difficult to engineer and expensive, says James Haber, a molecular biologist at Brandeis University in Waltham, Massachusetts. CRISPR works differently: it relies on an enzyme called Cas9 that uses a guide RNA molecule to home in on its target DNA, then edits the DNA to disrupt genes or insert desired sequences. Researchers often need to order only the RNA fragment; the other components can be bought off the shelf. Total cost: as little as $30. “That effectively democratized the technology so that everyone is using it,” says Haber. “It's a huge revolution.”
Now the warnings:
“This power is so easily accessible by labs — you don't need a very expensive piece of equipment and people don't need to get many years of training to do this,” says Stanley Qi, a systems biologist at Stanford University in California. “We should think carefully about how we are going to use that power.”...
“People just don't have the time to characterize some of the very basic parameters of the system,” says Bo Huang, a biophysicist at the University of California, San Francisco. “There is a mentality that as long as it works, we don't have to understand how or why it works.” That means that researchers occasionally run up against glitches. Huang and his lab struggled for two months to adapt CRISPR for use in imaging studies. He suspects that the delay would have been shorter had more been known about how to optimize the design of guide RNAs, a basic but important nuance.
...Doudna has begun to have more serious concerns about safety. Her worries began at a meeting in 2014, when she saw a postdoc present work in which a virus was engineered to carry the CRISPR components into mice. The mice breathed in the virus, allowing the CRISPR system to engineer mutations and create a model for human lung cancer4. Doudna got a chill; a minor mistake in the design of the guide RNA could result in a CRISPR that worked in human lungs as well. “It seemed incredibly scary that you might have students who were working with such a thing,” she says. “It's important for people to appreciate what this technology can do.”
Andrea Ventura, a cancer researcher at Memorial Sloan Kettering Cancer Center in New York and a lead author of the work, says that his lab carefully considered the safety implications: the guide sequences were designed to target genome regions that were unique to mice, and the virus was disabled such that it could not replicate. He agrees that it is important to anticipate even remote risks. “The guides are not designed to cut the human genome, but you never know,” he says. “It's not very likely, but it still needs to be considered.”
As the article later notes, it might end up being a case like the earlier excitement about gene therapy falling apart, when researchers discovered it was a lot trickier to administer that hoped, and could kill.
This seems very likely to me.
My hunch, expressed in an earlier post, was that working on the molecular genetic scale is never likely to be easy and would be readily capable of having unintended consequences on other bits of the gene. Seems I was right:
Yet many scientists caution that there is much to do before CRISPR can be deployed safely and efficiently. Scientists need to increase the efficiency of editing, but at the same time make sure that they do not introduce changes elsewhere in the genome that have consequences for health. “These enzymes will cut in places other than the places you have designed them to cut, and that has lots of implications,” says Haber. “If you're going to replace somebody's sickle-cell gene in a stem cell, you're going to be asked, 'Well, what other damage might you have done at other sites in the genome?'”What's more, I wouldn't be confident that even the successful removal of certain bits of DNA which cause disease might not turn out to have other, non desired, effects, but no one in the article addresses that.
Keith Joung, who studies gene editing at Massachusetts General Hospital in Boston, has been developing methods to hunt down Cas9's off-target cuts. He says that the frequency of such cuts varies widely from cell to cell and from one sequence to another: his lab and others have seen off-target sites with mutation frequencies ranging from 0.1% to more than 60%. Even low-frequency events could potentially be dangerous if they accelerate a cell's growth and lead to cancer, he says.
As the article goes on to also explain, the technique has the potential to bioengineering animals that always pass on the new characteristic, leading to the possibility of completely eradicating species very quickly. But at what ecological cost?
So libertarians can get as uptight as they like about bioethicists who are philosophically opposed to editing the human genome for permanent changes down the line, but they ought to look at the real and practical issues with the process because they get too excited about its potential.
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Have you ever clocked someone yawning in the office and soon followed suit? How about 'catching' a yawn on public transport? Well, now a new study from the University of Nottingham has revealed the part of our brain that triggers the yawning response.
According the research, it's the part of the brain that deals with motor function – the primary motor cortex, which also plays a part in conditions such as Tourette's syndrome.
Stephen Jackson, a professor of cognitive neuroscience and lead author, told Medical News Today why he thinks yawning is contagious:
""[...] there are many theories for why we yawn (e.g., lack of oxygen, to cool the brain, because we are tired, etc., etc.) but the evidence for these is lacking. The popular theory for contagious yawning is that it is linked with empathy for others, mimicry, and social bonding. Again the evidence for this is weak. I still think that much more research is required to understand the function and biology of yawning.
"…when I teach about yawning, I can get most of the class yawning. (Note, this doesn't happen for my other lectures)."
For the study, the researchers monitored 36 participants while they watched other people yawning. Some were told to stifle the urge. The urge was due to the "excitability" of the volunteer's primary motor cortex. Using a non-invasive transcranial magnetic stimulation (TMS) procedure it was also possible to increase this excitability and, as a result, more yawning.
Interestingly, the scientists believe that knowing about yawns and this part of the brain may offer some insight in to other disorders and unrelated conditions, such as echophenomena (also known as echo phenomenon that cause automatic imitations that occur without awareness), which is seen in Tourettes, epilepsy and autism.
Prof. Jackson adds:
"We suggest that these findings may be particularly important in understanding further the association between motor excitability and the occurrence of echophenomena in a wide range of clinical conditions that have been linked to increased cortical excitability and/or decreased physiological inhibition such as epilepsy, dementia, autism, and Tourette's syndrome."
He added the team are looking for potential non-drug, personalised treatments.
The research was published in the journal Current Biology.
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That the 200-kilometer (125-mile) Chicxulub crater in present-day Mexico formed when the asteroid struck, ultimately killing three quarters of life on the planet, is a fact that most scientists agree. But the trajectory and direction of this impact is still a matter of debate.
In a new study, an international team of researchers said their 3D simulations showed that the asteroid hit at an angle of 40 to 60 degrees – what Gareth Collins, a professor of planetary science at the science department of land and engineering of Imperial College London, described it as a worst case scenario for dinosaurs.
“We know this was one of the worst-case scenarios for impact mortality, because it put more dangerous debris in the upper atmosphere and scattered it everywhere – which led to a nuclear winter,” he added.
Such a strike has likely sparked billions of tons of sulfur and other gases into the atmosphere, blocking the sun and leading to a dramatic cooling of the Earth’s climate.
“This was based on a different interpretation of geophysical data, which our work overturns, and observations at the time that suggested that the ejecta from the crater was asymmetric, with more ejecta in North America (northwest) than elsewhere”, he explained via email, referring to the material that was expelled as a result of the impact.
“More recent observations have shown that the ejecta distribution is more or less symmetric.”
The team of researchers from Imperial College London, the University of Freiburg in Germany and the University of Texas in Austin examined the shape and structure of the crater and the rocks extracted by drilling the crater, which contained evidence of the extreme forces generated from the impact.
“Despite being buried under nearly a kilometer of sedimentary rocks, it is remarkable that geophysical data reveal so much about the structure of the crater – enough to describe the direction and angle of the impact,” said Auriol Rae, postdoctoral researcher at the University of Friborg and co-author of the study.
This information and other data were used to build a model that simulated the formation of the Chicxulub crater, determining the direction from which the asteroid came and the angle. The team considered four different angles: 90, 60, 45 and 30 degrees.
The authors said they considered 60 degrees to be the most likely angle due to the relationship between three points of the crater – its center, a mountain ring made of heavily fractured rock within the crater rim and the center of rocks of the dense and elevated mantle some 30 kilometers under the crater.
On the Chicxulub crater, these characteristics are aligned in the south-west-north-east direction, according to the study, and the team’s 3D simulations at a 60-degree angle reproduced these observations almost exactly.
The authors said that that angle of impact would have produced more climate-modifying gases such as sulfur and carbon dioxide than a very superficial or almost vertical impact.
Coffee enthusiast. Travel scholar. Infuriatingly humble zombie fanatic. Thinker. Professional twitter evangelist.
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Human history in Florida is replete with flooding experiences, most notably the devastating hurricanes of the early 20th century that brought about the systems of canals and dikes that make much of the central and south Florida habitable. It is not surprising that flooding frequently occurs in a state that originally was one-half wetlands. Without proper land use controls, flood protection measures can actually contribute to increased flood risks by creating a false sense of security which encourages unwise development in areas subject to flooding. Also, as new development further modifies stormwater runoff patterns, flood risks are often increased for areas that were not previously flood prone.
Flooding can occur in either floodplains (low-lying lands around rivers and streams, lakes, and wetlands), or in other low-lying, poorly drained areas. Flooding occurs when rainfall is too intensive for the land to absorb the extra runoff, when natural or artificial waterways are inadequate to accommodate runoff. The Federal Emergency Management Agency (FEMA) estimates about 14.25 million acres, or 41 percent, of Florida is flood prone -- the highest percentage of all 50 states. The Department of Community Affairs (DCA) estimates that about 1.3 million people live in areas subject to flooding.
Flooding in Florida typically is caused by heavy or prolonged rainfall from tropical storms and hurricanes. Rainfall in Alabama and Georgia can cause significant flooding problems in North Florida as shown during tropical storms Alberto and Beryl in 1994. Florida's high vulnerability to flooding demands an adequate response to protect the public health, safety, and welfare. The economic and social impacts of flooding events can be staggering. For instance, statewide damage from three tropical storms and two tropical disturbances in 1993 was approximately 500 million dollars. Future public liabilities related to flood losses can be greatly reduced through proper control of development in floodplains and floodprone areas and maintenance of the existing flood protection infrastructure.
Human occupancy of and alteration of floodplains and floodprone areas are threatening public safety, health, and welfare.
The cornerstone of any floodplain management strategy is adequate mapping of floodplains and flood prone areas. However, because floodplain mapping is a complex, expensive, and time-consuming endeavor, many areas are not adequately mapped. Many floodplain mapping efforts have occurred in response to specific needs in specific areas, but a coordinated, comprehensive approach has not been undertaken. Adequate mapping is an important link between landuse and water resources planning.
For the subject area of the flooding, the best current formal recognition of such areas is the 10- and 100-year floodplains mapped by various agencies. Protecting the functions of unaltered floodplains is a critical aspect of statewide ecosystem
management efforts. As many floodplain areas have been altered, restoring their natural functions is also an important issue. Land acquisition and management through various programs provides a very effective tool for protecting and restoring floodplains.
Floodplain management responsibilities are shared among federal, state, regional, and local governments. Local governments have the most direct control in floodplain management through landuse planning and regulation, land acquisition and management, and as sponsors for the flood insurance program administered by FEMA. Water Management Districts (WMDs) and the DEP, through surface water management regulations also regulate development activities in floodplains and flood prone areas.
Inadequate preparation for flood disasters and response have increased property damage and risks to human safety.
The State of Florida Comprehensive Emergency Management Plan, administered by the DCA, Division of Emergency Management, coordinates the activities and responsibilities of 23 state agencies, 5 WMDs, school districts, and numerous private organizations during declared emergencies. The experience of Hurricane Andrew in 1992 sharpened the state's awareness of the need to be prepared for, and respond to, flooding and other natural disasters.
The most effective opportunity to improve emergency management procedures, however, is after emergency situations occur and emergency procedures are completed. An ongoing procedure to evaluate the effectiveness of emergency management procedures, after the emergency has passed, needs to be coordinated among all responsible entities.
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Regulating Earth's climate with micro-organisms
Scientists have sought to learn more about how the Earth's oceans absorb carbon dioxide and generally exchange gases with the atmosphere so they can better understand the corresponding effects on climate. To that end, many researchers are turning their attention to the microscopic organisms that help recycle carbon, nitrogen, sulfur and other elements through the oceans. Finding out exactly how and to what degree they do that is an ongoing scientific challenge, and scientists may first have to learn more about how the microbes interact with their environment at the scale of the individual microbe.
In recent work, an international team of scientists led by Professor Roman Stocker of the MIT Department of Civil and Environmental Engineering opened a window into that microbial world. The team studied how certain strains of marine microbes find and use sulfur, an element vital to many of them. Some microbes ingest the sulfur, convert it and pass it back into the ocean in altered form, keeping the chemical moving through Earths sulfur cycle.
Using video microscopy, the scientists captured digital images of the single-celled microbes swimming toward two forms of sulfur: dimethylsulfide (DMS), the chemical responsible for the slightly sulfuric smell of the sea, and its precursor dimethylsulfoniopropionate (DMSP), which can be converted to DMS by the microbes. DMS is known to influence climate; when it moves from the ocean to the atmosphere as a gas, it oxidizes, forming cloud condensation nuclei which promote cloud formation over the ocean. These clouds reflect sunlight rather than allowing it to heat the Earths surface.
Stocker, Justin Seymour, a former postdoctoral fellow at MIT who is now a research fellow at the University of Technology Sydney, Professor Rafel Simó of the Institute for Ocean Sciences in Barcelona, and MIT graduate student Tanvir Ahmed reported this research which was funded by the Australian Research Council, the Spanish Ministry of Science and Innovation, La Cambra de Barcelona, the Hayashi Fund at MIT, and the National Science Foundation earlier this year in the journal Science.
It had been previously demonstrated that DMSP and DMS draw coral reef fish, sea birds, sea urchins, penguins and seals, suggesting that these chemicals play a prominent ecological role in the ocean. Now we know that they also attract microbes, said Stocker. But this is not simply adding a few more organisms to that list. The billions of microbes in each liter of seawater play a more important role in the oceans chemical cycles than any of the larger organisms.
Stocker has pioneered the use of microfluidic technology to study the behavior of marine microbes in the laboratory. He re-creates a microcosm of the ocean environment using a device about the size of a flash drive, made of clear rubbery material engraved with minuscule channels into which he injects ocean water, microbes and food in the form of dissolved organic matter. Then, using a camera attached to a microscope, he records the microbes response. In the past few years, he has recorded microbes as they use their whip-like flagella to swim toward food, a finding that contradicts the traditional view of marine microbes as passive feeders.
In the latest research, the scientists injected different chemicals into the channels of the device in a way that mimicked the bursting of a microbial cell after a viral infection a common event in the ocean. Although they performed the tests using several substances, including DMS, the scientists focused primarily on DMSP, which is produced by some phytoplankton and released into the water when a cell explodes. That DMSP can dissolve in the water or be transformed by other microbes into DMS, which also dissolves in the water before being released as a gas into the atmosphere.
The research indicates that the chemicals odor does draw microbial predators, much as its smelly cousin DMS does at larger scales. This is the first such study to make a visual record of microbial behavior in the presence of DMSP.
The team selected seven microbial species that are roughly analogous to plants, herbivores and predators in the animal kingdom: three photosynthetic microbes (phytoplankton), two heterotrophic bacteria that feed off the carbon produced by other microbes, and two microzooplankton that prey on other microbes.
Six of the seven microbial species tested were attracted to the DMSP in the microfluidic device; only one species a phytoplankton ignored it. Some of the species displayed the strongest swimming responses among any of the 100 or so cases yet tested by Stocker and Seymour in their research projects. This, Stocker said, is a clear indication that DMSP acts as a powerful chemical cue.
The researchers also found that some marine microbes, including bacteria, are attracted to DMSP because they feed on it, while others, the microzooplankton, are drawn to the chemical because it signals the presence of prey. This challenges previous theories that DMSP might deter predators. Our observations clearly show that, for some plankton, DMSP acts as an attractant towards prey rather than a deterrent, said Simó.
Farooq Azam of the Scripps Institution of Oceanography, one of the first scientists to recognize the importance of microbes in the ocean food chain, agrees. The findings of this study are exciting and unexpected in showing how broadly distributed throughout the microbial food web is the ability to sense DMSP and to behaviorally respond to it. In view of the significance of DMSP and DMS in global climate, these results should stimulate future research to understand how the potentially complex microbial interactions are reflected in the regulation of the fluxes of DMS and DMSP.
Azam also said that the study adds substantial weight to the emergent view that understanding how microbes control the grand cycles of elements in the ocean and global climate will require study at the scale of the individual microbe.
The researchers are now working on a system to replicate their experiments on oceanographic ships using bacteria collected directly from the ocean, rather than lab-cultured microbes. This will allow them to use microfluidics to create a virtual microbe aquarium at sea.
Were doing for microbes what ecologists have done with larger organisms for a long time, said Stocker. Were observing them in order to better understand their behavior.
This story is republished courtesy of MIT News (web.mit.edu/newsoffice/), a popular site that covers news about MIT research, innovation and teaching.
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A furnace is a device in which heat is generated and transferred to materials with the object of bringing about physical and chemical changes. The source of heat is usually combustion of solid, liquid or gaseous fuel, or electrical energy applied through resistance heating (Joule heating) or inductive heating. However, solar energy can provide a clean source of high temperature if focused onto a small area. This was recognized over two hundred years ago by Lavoisier who built a large mobile "magnifying glass" system, Figure 1, to bring about the combustion of metals in a sealed glass container and subsequently the demise of the phlogiston theory.
Furnaces employing combustion produce a hot gas which transfers heat to the material by radiation and convection. Solids are heated by direct contact, but fluids are usually heated indirectly, being carried inside pipes within the furnace. Alternatively, a Regenerative Heat Exchanger may be used to transfer heat from the combustion gases. Indirect heating has the advantage of avoiding contamination by combustion products.
There are two principle categories of indirect heating furnace. The first is that of Boilers, where the heat is used to generate steam for power generation or process plant use. The second is that of furnaces designed to heat process fluids other than water. Some of the latter category are conventional heaters used simply to increase the fluid temperature, others are process heaters used to bring about physical and chemical changes in the products, for example distillation, and pyrolysis of hydrocarbons or catalytic steam-gas reforming of synthetic natural gas.
In the petrochemical industry process furnaces, or process heaters, are usually fired by oil or gas. They are designed to ensure that the fluid receives the right amount of heat and has the required residence time within specified temperature limits. This is achieved by appropriate disposition of tubes and careful control of firing rate and fluid flow. The following Figure 2 shows three typical geometries.
The following tables give the properties of gaseous, liquid and solid fuels used in furnaces. Gross calorific value includes the heat released by the condensation and cooling of combustion products, and the combustion air/fuel ratio is the stoichiometric value based on theoretically perfect combustion (see Combustion). In practice, excess air is required to ensure complete combustion. The excess is usually about 10% of the stoichiometric value for premixed gaseous fuels, 20% for distillate oil and pulverized coal, and 30% for residual oil, although lower values may be achieved with efficient burners.
The rate of heat released in a fuel fired furnace is given by the product of the mass rate of feed of fuel, , and the calorific value Δhf, i.e.,
The heat generated by combustion appears initially as sensible heat in the gaseous products of combustion, generated at the rate . If this were an adiabatic process (i.e., no heat transfer), the gas would attain the adiabatic flame temperature, Tf, given by
where T0 is the inlet air/fuel temperature and cpg an appropriate averaged specific heat capacity of the gases for the range T0 to Tf.
However, as illustrated in Figure 3, part of the heat generated passes to the tubes containing the process fluid at rate and part is lost through the furnace walls at rate . The remainder escapes at rate as waste heat of the combustion products leaving the furnace. The overall heat balance therefore is
Because of this the actual gas temperature, Tg, reached in the furnace is less than the adiabatic flame temperature, Tf. Neglecting wall losses an approximate value of Tg can be obtained from
For example, in a typical oil fired furnace the value of Tf is 2,200 K and the value of Tg is 1,300 K [Shires, Hewitt and Bott (1994)].
Table 2. Properties of selected liquid and solid fuelsa
|(aFrom the Heat Exchanger Design Handbook, Hemisphere Publications. With permission.)|
The rate of heat transfer, , from the hot gases to the tubes is a function of the gas temperature as well as the radiation properties of the gases, the geometry of the furnace and tubes, and the emissivity of the tube surface. In applying simple furnace models iteration is therefore necessary.
In most process heaters, the major part of the heat transfer from the hot gases to the tubes is by radiation. To calculate the radiative component it is necessary to know the effective emissivity, εg, of the combustion gases (typical value 0.25). This is dependent on the ratio of the partial pressures of CO2 and H2, the temperatures of the gas and the radiation source and the effective size of the radiating gas cloud. The latter is represented by the term pLo, the product of partial pressure and effective length of the furnace—a term first introduced by H. C. Hottel. For details of the procedure see Hottel and Sarofim (1967) and Hewitt, Shires and Bott (1994).
The rate of heat transfer to the furnace product is also a function of the geometry of the tube banks and the fraction of the furnace surface area covered by them.
If the effective area of the tubes, A1, is only a small fraction of the total area. At, (Figure 4a), the surface receives blackbody radiation (see Radiative Heat Transfer) and the rate of heat transfer is independent of gas emissivity, εg, i.e.,
where ε1 is the effective emissivity of the receiving surface and T1 its temperature and σ is the Stefan-Boltzmann constant, 5.670 × 10−8 W/m2K4. On the other hand, if A1 is almost equal to the total surface area, At, then
Intermediate ratios are covered by the speckled surface equation [Hottel and Sarofim (1967)].
where C = A1/At.
In practice the area, A1, receiving radiation is a tube bank, Figure 5, which intercepts only a fraction of the incident radiation; some passes through to the refractory wall and is re-irradiated. For a detailed analysis see Hottel and Sarofim (1967).
The fraction of radiation intercepted by a single row of tubes having a pitch to diameter ratio B (equals p/d) is given by
Figure 6 shows F as a function of B for one and two rows of tubes.
Figure 6. Fraction of incident radiation intercepted by tubes. (From the Heat Exchanger Design Handbook, Hemisphere Publications. With permission.)
The effective emissivity based on the projected area using Eq. (7) and allowing for re-irradiation from the refractory backing is
This is illustrated in Figure 7 for a typical tube surface emissivity of 0.85.
The relationship between the rate of heat transfer, , from the gas to the tubes and the gas temperature, Tg, is obtained by combining Eqs. (6), (7) and (8). This in turn can be combined with Eq. (3) to determine the furnace thermal characteristics. An example of this procedure is described in Hewitt, Shires and Bolt (1994).
The complete mathematical description of a practical furnace is complex, combining aerodynamics, chemical reactions and heat transfer, and computer programs are normally used for detailed solutions. There are two basic types of approach; zone methods and flux methods.
Zone methods are employed when the heat release pattern from the flame is known or can be calculated independently. Conceptually, the furnace and its walls are divided into discrete zones, the effective exchange areas between zones are determined, and radiative heat transfer corresponding to the prescribed heat release pattern is calculated.
In flux methods, instead of dividing the space into zones the radiation arriving at a point in the system is itself divided into a number of characteristic directions, representing averages over a specified solid angle. Flux methods are well suited for use in combination with modern methods of prediction of fluid flow and mixing. Simultaneous solutions of the radiative heat transfer equations using flux methods and turbulent flow models are feasible.
For further information on these two methods see Beer (1974), and Afgan and Beer (1974), where examples of their application can be found.
As a first approach to the estimation of furnace performance the well stirred furnace model is relatively simple and quick to use. One of the first versions was introduced by Lobo and Evans (1939), and was used by Kern (1986). An improved version expressed in nondimensional terms was introduced by Hottel and Sarofim (1967). This was reveiwed by Hottel (1974) and subsequently described by Truelove (1983), and Hewitt, Shires and Bott (1994).
The general performance equation for furnaces derived from this model is
where d is a factor to account for imperfect mixing, approximately equal to 1.2, Te is external temperature, Ur is overall heat transfer coefficient for the refractory wall, Ar is the area of the refractory surface, A1 is the area of tube subject to radiative heat transfer, Ac is the area of tube subject to convective heat transfer and α the average convective heat transfer coefficient for Ac.
Figure 8a shows graphically the well stirred furnace model performance prediction for zero wall losses, and Figure 8b for the same conditions but with typical wall losses included. Comparison of these figures shows that wall losses have a very significant effect when tube temperatures are high. As the firing rate, represented by D'd, is reduced, the efficiency, represented by , reaches a peak then falls, eventually approaching zero as the major part of the heat is lost through the walls.
Figure 8. Performance curves for stirred reactor furnace with negligible wall losses. (From the Heat Exchanger Design Handbook, Hemisphere Publications. With permission.)
Figure 9. Performance curves for stirred reactor furnace with wall losses; , . (From the Heat Exchanger Design Handbook, Hemisphere Publications. With permission.)
Afgan, N. H. and Beer, J. M. (1974) Heat Transfer in Flames, Scripta Book Co., Wiley, New York.
Beer, J. M. (1974) Methods of calculating radiative heat transfer from flames in combustors and furnaces, Heat Transfer in Flames, Scripta Book Co., Wiley, New York.
Hewitt, G. F. Shires, O. L., and Bott T. R. (1994) Process Heat Transfer, CRC Press.
Hottel, H. C. (1974) First estimates of industrial performance; the one—gas—zone model re-examined, Heat Transfer in Flames, Scripta Book Co., Wiley, New York.
Hottel, H. C. and Sarofim, A. F. (1967) Radiative Heat Transfer, McGraw-Hill, New York.
Kern, D. O. (1986) Process Heat Transfer, McGraw-Hill, New York.
Lobo, W. E. and Evans, J. E. (1939) Trans AIChE, 35-743.
Truelove, J. S. (1983) Furnaces and combustion chambers. Heat Exchanger Design Handbook, Hemisphere Publishing, New York.
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Resources tagged with: Creating and manipulating expressions and formulae
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There are 124 results
Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae
Number Pyramids
Age 11 to 14 Challenge Level:
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Partitioning Revisited
Age 11 to 14 Challenge Level:
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Interactive Number Patterns
Age 14 to 16 Challenge Level:
How good are you at finding the formula for a number pattern ?
More Number Pyramids
Age 11 to 14 Challenge Level:
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Magic W
Age 14 to 16 Challenge Level:
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
Summing Consecutive Numbers
Age 11 to 14 Challenge Level:
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Christmas Chocolates
Age 11 to 14 Challenge Level:
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Multiply the Addition Square
Age 11 to 14 Challenge Level:
If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?
A Tilted Square
Age 14 to 16 Challenge Level:
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Painted Cube
Age 14 to 16 Challenge Level:
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
Special Numbers
Age 11 to 14 Challenge Level:
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Always the Same
Age 11 to 14 Challenge Level:
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
Partly Painted Cube
Age 14 to 16 Challenge Level:
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Attractive Tablecloths
Age 14 to 16 Challenge Level:
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Top-heavy Pyramids
Age 11 to 14 Challenge Level:
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Multiplication Square
Age 14 to 16 Challenge Level:
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
AMGM
Age 14 to 16 Challenge Level:
Can you use the diagram to prove the AM-GM inequality?
Special Sums and Products
Age 11 to 14 Challenge Level:
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Pair Products
Age 14 to 16 Challenge Level:
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Cubes Within Cubes Revisited
Age 11 to 14 Challenge Level:
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
Regular Hexagon Loops
Age 11 to 14 Challenge Level:
Make some loops out of regular hexagons. What rules can you discover?
Quick Times
Age 11 to 14 Challenge Level:
32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.
Janine's Conjecture
Age 14 to 16 Challenge Level:
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
Steel Cables
Age 14 to 16 Challenge Level:
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
The Number Jumbler
Age 7 to 14 Challenge Level:
The Number Jumbler can always work out your chosen symbol. Can you work out how?
Magic Squares for Special Occasions
Age 11 to 16
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
Chocolate 2010
Age 14 to 16 Challenge Level:
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
How Much Can We Spend?
Age 11 to 14 Challenge Level:
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
Age 11 to 14 Challenge Level:
Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .
Age 11 to 14 Challenge Level:
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Crossed Ends
Age 11 to 14 Challenge Level:
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Matchless
Age 14 to 16 Challenge Level:
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
Seven Squares
Age 11 to 14 Challenge Level:
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Lower Bound
Age 14 to 16 Challenge Level:
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Always a Multiple?
Age 11 to 14 Challenge Level:
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Sum Equals Product
Age 11 to 14 Challenge Level:
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Triangles Within Pentagons
Age 14 to 16 Challenge Level:
Show that all pentagonal numbers are one third of a triangular number.
Marbles in a Box
Age 11 to 16 Challenge Level:
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Chocolate Maths
Age 11 to 14 Challenge Level:
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Triangles Within Squares
Age 14 to 16 Challenge Level:
Can you find a rule which relates triangular numbers to square numbers?
Triangles Within Triangles
Age 14 to 16 Challenge Level:
Can you find a rule which connects consecutive triangular numbers?
Odd Differences
Age 14 to 16 Challenge Level:
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nĀ² Use the diagram to show that any odd number is the difference of two squares.
Pareq Calc
Age 14 to 16 Challenge Level:
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Even So
Age 11 to 14 Challenge Level:
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
Age 11 to 14 Challenge Level:
Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .
What's Possible?
Age 14 to 16 Challenge Level:
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Age 14 to 16 Challenge Level:
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
Age 11 to 14 Challenge Level:
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
Sums of Pairs
Age 11 to 16 Challenge Level:
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
Generating Triples
Age 14 to 16 Challenge Level:
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
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This image of Saturn was taken by Voyager I in 1980
Click on image for full size
Courtesy of NASA
Like the inner planets and Jupiter, Saturn is clearly visible in the night sky. The ancient Greeks named the planet after the god of agriculture and time. It wasn't until 1655, however, that we knew Saturn had rings. Galileo saw two lumps on either side of Saturn, be he didn't know what they were. The astronomer Christian Huygens later found out they were rings.
Much of what we now know about the second largest planet in our solar system comes from the Voyager spacecrafts. Voyager took close-up pictures of Saturn and its rings. They clearly show the large gaps in between the rings, called the Cassini Division and Encke Division, after the two scientists that supposedly discovered them.
Recently, a lot of research has been devoted to Saturn's moons. Huygens discovered Saturn's largest moon, Titan, in 1655. Voyager later showed that Titan actually has an atmosphere. Mimas, Enceladus, Tethys, Dione, Rhea and Iapetus were discovered by various astronomers. Many smaller moons were found by the Voyager spacecraft.
Cassini is the newest mission to Saturn. If all goes as planned, it will reach the planet in 2004. Cassini will study Saturn and its largest moon, Titan.
You might also be interested in:
The rare geometric arrangement of planets Jupiter, Saturn, Uranus, and Neptune in the 1980's made it possible for the Voyager spacecrafts to visit them over a 12 year span instead of the normal 30. They...more
Like the inner planets and Jupiter, Saturn is clearly visible in the night sky. The ancient Greeks named the planet after the god of agriculture and time. It wasn't until 1655, however, that we knew Saturn...more
The dramatic appearance of Saturn stems mainly from the spectacular rings. What is visible of the atmosphere is much less dramatic. The clouds of Saturn are much less colorful than those of Jupiter. This...more
The Giant planets do not have the same layered structure that the terrestrial planets do. Their evolution was quite different than that of the terrestrial planets, and they have less solid material inside....more
Many people are fascinated by Saturn's rings. Although Saturn isn't the only planet with rings, it is the only planet famous for them. Almost every image or drawing of the planet has the rings included....more
Saturn's magnetosphere is not as big as Jupiter's, but is very large nonetheless. It extends well beyond the orbits of Saturn's moons. It is probably generated in the same manner as is Jupiter's, which...more
There's a lot of strange and interesting stuff going on at both the North and South Poles of Saturn. Features at the poles of two of Saturn's moons, Titan and Enceladus, have also grabbed the attention...more
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On this day (Jan. 27) in 1967, NASA astronauts Virgil “Gus” Grissom, Ed White and Roger Chaffee died in a pad fire inside of the Apollo 1 spacecraft that was supposed to lift off only a month hence. The tragedy shocked NASA, which was then aiming for manned landings on the moon, and caused an in-depth investigation into the spacecraft’s construction and the cause of the fire.
Above, you can see one of the first news reports after the fire took place, from ABC’s Jules Bergman and a correspondent at “Cape Kennedy” (which is called Cape Canaveral today, referring to an area adjacent to the Kennedy Space Center where the launch was supposed to take place.) “It was too late from the beginning,” Bergman said in the report, referring to the frantic effort to get the astronauts out of their burning spacecraft.
An investigation determined that a spark flew from somewhere inside of the spacecraft and easily ignited in the pure-oxygen atmosphere, fuelled by fire-friendly materials inside the spacecraft. The astronauts were unable to get out quickly because the hatch was complicated to open. The redesigned Apollo spacecraft featured a swift-to-open hatch, fewer flammable materials, covered electrical connections (to mitigate against short-circuits), and a mixed atmosphere of oxygen and nitrogen on the ground.
Safety measures arising from the tragedy did help with saving astronauts on other flights, notably Apollo 13. That mission saw an oxygen tank explode en route to the moon in April 1970.
Every year, NASA has a day of remembrance to commemorate lost crews. The Apollo 1 anniversary marks a solemn week in the agency, as it comes one day before the anniversary of the 1986 Challenger explosion that killed seven astronauts (Jan. 28) and a few days before the 2003 anniversary of the Columbia shuttle breakup, which killed another seven people (Feb. 1).
Four cosmonauts have died during spaceflight, all upon re-entry: Vladimir Komarov (during Soyuz 1 on April 24, 1967) and Georgi Dobrovolskiy, Viktor Patsayev, and Vladislav Volkov (during Soyuz 11 on June 30, 1971).
Training accidents have also claimed a few lives; a list of American ones is maintained at the Astronaut Memorial Foundation.
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The aorist tense "presents an occurrence in summary, viewed as a whole from the outside, without regard for the internal make-up of the occurrence."1 Wallace explains,
This contrasts with the present and imperfect, which portray the
action as an ongoing process. It may be helpful to think of the aorist
as taking a snapshot of the action while the imperfect (like the
present) takes a motion picture, portraying the action as it unfolds.
[A footnote points out,] There is a difference between seeing the
aorist as undefined and seeing it as a summary tense, though the two
are closely related. In our view the aorist summarizes. It is thus not
undefined or unmarked. That is to say, it is not necessarily the
“default” tense that one would use unless he or she had reason to use
another. The key issue, it seems, is the tense-mood combination.
Outside the indicative, the aorist is hardly unmarked (statistically,
the present runs neck-and-neck with it). However, in the indicative,
the aorist does appear to function this way, at least in narrative
literature. The imperfect, (historical) present, perfect, and
pluperfect are all used in narrative, along with the aorist. But the
aorist is by far the most common. Thus, the analogy with a snapshot
seems appropriate, enabling the student to get a handle on the basic
notion of the aorist’s aspect.2
Wallace goes on to share a helpful analogy:
Suppose I were to take a snapshot of a student studying for a mid-term
exam in intermediate Greek. Below the picture I put the caption,
“Horatio Glutchstomach studied for the mid-term.” From the snapshot
and the caption all that one would be able to state positively is that
Horatio Glutchstomach studied for the mid-term. Now in the picture you
notice that Horatio has his Greek text opened before him. From this,
you cannot say, “Because the picture is a snapshot rather than a
movie, I know that Horatio Glutchstomach only had his Greek text
opened for a split-second”! This might be true, but the snapshot does
not tell you this. All you really know is that the student had his
Greek text open. An event happened. From the picture you cannot tell
for how long he had his text open. You cannot tell whether he studied
for four hours straight (durative), or for eight hours, taking a ten
minute break every 20 minutes (iterative). You cannot tell whether he
studied successfully so as to pass the test, or whether he studied
unsuccessfully. The snapshot does not tell you any of this. The
snapshot by itself cannot tell if the action was momentary,
“once-for-all”, repeated, at regularly recurring intervals, or over a
long period of time. It is obvious from this crude illustration that
it would be silly to say that since I took a snapshot of Horatio
studying, rather than a movie, he must have studied only for a very
Wallace further elaborates on the aorist in the indicative mood:
In the indicative, the aorist usually indicates past time with
reference to the time of speaking (thus, “absolute time”). Aorist
participles usually suggest antecedent time to that of the main verb
(i.e., past time in a relative sense). There are exceptions to this
general principle, of course, but they are due to intrusions from
other linguistic features vying for control....
Outside the indicative and participle, time is not a feature of the
In conclusion, we must avoid the dangers of saying too much or too little when the aorist tense occurs. The aorist doesn't exist "in a vacuum," so we must pay attention to when context is guiding us towards different meanings as it combines various linguistic features. But we must also avoid saying too much (i.e. saying the aorist always means "once-for-all" or "momentary" action).
1 Fanning, Verbal Aspect, 97. Cf. also McKay, “Time and Aspect,” 225.
2 Daniel B. Wallace, Greek Grammar Beyond the Basics: An Exegetical Syntax of the New Testament (Grand Rapids, MI: Zondervan, 1996), 555.
3 Ibid., 555.
4 Ibid., 555.
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As compared to Macbeth in Shakespeare's Macbeth, who was the real Macbeth in Scotland's history?
1 Answer | Add Yours
Shakespeare was an expert in his craft and was keenly aware what his audiences expected of him and how to ensure his own popularity. Macbeth was written in consideration of James I for whom Shakespeare staged his version of Macbeth and taking much of his information from the Holingshed Chronicles of England. James I was believed to be related to Banquo and in Shakespeare's Macbeth, Banquo is honorable and loyal to his king.
Mac Bethad mac Findláich, translated into English as Macbeth, was born in about 1005. His father was an earl of Moray.
Macbeth did indeed succeed Duncan to the throne in Scotland , in the eleventh century. Duncan was the grandson of Malcolm II to whom Macbeth was related on his mother's side. Furthermore, Duncan was killed by Macbeth during a raid on Moray. Macbeth became King of Scotland and Duncan's wife fled with her sons Malcolm III and Donald. There is controversy over the actual recording of historical events and, as expected, different versions.
Macbeth is believed to have successfully reigned for some years. Some claim that Macbeth was a tyrant and there is confusion over names that were attributed to him - or whether these names actually belonged to other people with whom he was associated. He was defeated during a successful attempt to return Duncan's son Malcolm III to the throne.
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Suzanne E. Smith’s research into African American funeral traditions is featured in the 2013 documentary Homegoings:
From antebellum slavery to the twenty-first century, African American funeral directors have orchestrated funerals or “homegoing” ceremonies with dignity and pageantry. As entrepreneurs in a largely segregated trade, they were among the few black individuals in any community who were economically independent and not beholden to the local white power structure. Most important, their financial freedom gave them the ability to support the struggle for civil rights and, indeed, to serve the living as well as bury the dead.
During the Jim Crow era, black funeral directors relied on racial segregation to secure their foothold in America’s capitalist marketplace. With the dawning of the civil rights age, these entrepreneurs were drawn into the movement to integrate American society, but were also uncertain how racial integration would affect their business success. From the beginning, this tension between personal gain and community service shaped the history of African American funeral directing.
For African Americans, death was never simply the end of life, and funerals were not just places to mourn. In the “hush harbors” of the slave quarters, African Americans first used funerals to bury their dead and to plan a path to freedom. Similarly, throughout the long—and often violent—struggle for racial equality in the twentieth century, funeral directors aided the cause by honoring the dead while supporting the living. To Serve the Living offers a fascinating history of how African American funeral directors have been integral to the fight for freedom.
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Today we often hear that children must learn how to code as the only way to be prepared for the technical challenges they will encounter as adults. I used to teach coding to children at my school, but after reading Code Girls, I am reminded that we should not lose sight of the importance of a broad education that emphasizes languages, math, science, music, and all the other subjects that make up the liberal arts. This type of education serves people well, because it ensures that students possess thinking skills and develop the wherewithal to learn, when necessary, new types of technical skills, such as coding and programming.
in Code Girls author Liza Mundy tells the story of how young women, many of whom were academically prepared in the liberal arts to become teachers, came to work for the Army and Navy in Washington, D.C. during World War II. They mastered extraordinary technical skills, learning how to decipher complicated and perplexing coded wartime communications.
The book describes how thousands of these young women — recruited with no idea of what they would be doing — learned how figure out cyphers and break codes, thereby ensuring that the United States military knew what its enemies were planning or doing. The women poured over intercepted messages from enemy countries and by breaking open these messages helped to save lives, sink ships, down airplanes, and fool enemies. Their work helped to change the course for the United States in World War II. Continue reading
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# Spring 2 Week 1 24.2.23
Hello!
This week we have welcomed a very exciting addition to our reception classroom: an incubator complete with eight eggs! The children began the week by writing predictions about what might be inside the eggs. There were some very interesting ideas! On Wednesday afternoon the first of the eggs hatched! The children were very excited to meet the first chick. By today we have 3 chicks that have hatched out of 8 eggs. Unfortunately, the rest of the eggs will not hatch now but we look forward to taking care of the chicks we have hatched! The children have enjoyed writing about, drawing and painting pictures of the chicks.
We have been learning all about life cycles this week. We have learnt about the different stages of a human life then we also looked at a frog’s life cycle and a chicken’s life cycle. The children have seen the eggs hatching which has been really exciting but they were also able to see the mummies and daddies of the eggs when we had a zoom call with Debs from Eggucation on Thursday morning. She showed us around the farm where the eggs came from, we watched her feeding the chickens and we saw some more eggs in the coops. She then opened up the floor for questions and the children thought of some really interesting questions. They listened beautifully and watched with interest.
In maths we have been exploring the numbers 9 and 10, looking at how these numbers are represented in different ways. We have been looking at how the numbers 9 and 10 are made up. You might like to get two plates and 9 pieces of food (strawberries, apple pieces…) – how many different ways can you make 9? 1 and 8, 2 and 7 etc. Your child might like to take photos of all the different combinations. To find combinations of 10, you could string macaroni or penne pasta to some string or wool. If you string 10 pieces, then your child can move these along the string (some at one end and some at the other) to make different combinations. We have also been ordering numerals. The cheeky kangaroo played some mischievous games, mixing up numbers and taking numbers away and it was our job to find out what he’d done and how we could fix it. You might like to play this game at home. Make 1-10 number cards and put them into order. Then close your eyes while the cheeky kangaroo does something to the cards. Open your eyes. Can you fix it?
Next week we will be learning the story of Jasper’s Beanstalk for our new Talk 4 Writing unit. The children will be learning the story, using actions and then creating story maps to help retell the story.
In maths we will be comparing numbers within 10 and we will be continuing to make 10 in different ways, exploring number bonds.
Don’t forget to check our class page on the website for photographs of what we’ve been up to!
We hope you have a lovely weekend,
The Reception team
Top
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# Search by Topic
#### Resources tagged with Interactivities similar to Trice:
Filter by: Content type:
Stage:
Challenge level:
### There are 155 results
Broad Topics > Information and Communications Technology > Interactivities
### Muggles Magic
##### Stage: 3 Challenge Level:
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.
### Diagonal Dodge
##### Stage: 2 and 3 Challenge Level:
A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.
### Volume of a Pyramid and a Cone
##### Stage: 3
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
### Cogs
##### Stage: 3 Challenge Level:
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
### When Will You Pay Me? Say the Bells of Old Bailey
##### Stage: 3 Challenge Level:
Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?
### Khun Phaen Escapes to Freedom
##### Stage: 3 Challenge Level:
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
### Online
##### Stage: 2 and 3 Challenge Level:
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
### Picturing Triangle Numbers
##### Stage: 3 Challenge Level:
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
### Shear Magic
##### Stage: 3 Challenge Level:
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
### Tilted Squares
##### Stage: 3 Challenge Level:
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
### Rolling Around
##### Stage: 3 Challenge Level:
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
### Conway's Chequerboard Army
##### Stage: 3 Challenge Level:
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
### Poly-puzzle
##### Stage: 3 Challenge Level:
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
### You Owe Me Five Farthings, Say the Bells of St Martin's
##### Stage: 3 Challenge Level:
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
### Squaring the Circle and Circling the Square
##### Stage: 4 Challenge Level:
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
### Cubic Net
##### Stage: 4 and 5 Challenge Level:
This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!
### Square Coordinates
##### Stage: 3 Challenge Level:
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
### Sliding Puzzle
##### Stage: 1, 2, 3 and 4 Challenge Level:
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
### Isosceles Triangles
##### Stage: 3 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
### Disappearing Square
##### Stage: 3 Challenge Level:
Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .
### A Tilted Square
##### Stage: 4 Challenge Level:
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
### Jam
##### Stage: 4 Challenge Level:
To avoid losing think of another very well known game where the patterns of play are similar.
### Icosian Game
##### Stage: 3 Challenge Level:
This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.
### Diamond Mine
##### Stage: 3 Challenge Level:
Practise your diamond mining skills and your x,y coordination in this homage to Pacman.
### Shuffles Tutorials
##### Stage: 3 Challenge Level:
Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.
### Rollin' Rollin' Rollin'
##### Stage: 3 Challenge Level:
Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?
### Bow Tie
##### Stage: 3 Challenge Level:
Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.
### Interactive Spinners
##### Stage: 3 Challenge Level:
This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.
### Lost
##### Stage: 3 Challenge Level:
Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.
### Overlap
##### Stage: 3 Challenge Level:
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
### Fifteen
##### Stage: 2 and 3 Challenge Level:
Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.
### Inside Out
##### Stage: 4 Challenge Level:
There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .
### Top Coach
##### Stage: 3 Challenge Level:
Carry out some time trials and gather some data to help you decide on the best training regime for your rowing crew.
### Balancing 3
##### Stage: 3 Challenge Level:
Mo has left, but Meg is still experimenting. Use the interactivity to help you find out how she can alter her pouch of marbles and still keep the two pouches balanced.
### Factor Lines
##### Stage: 2 and 3 Challenge Level:
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
### Colour in the Square
##### Stage: 2, 3 and 4 Challenge Level:
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
### Tilting Triangles
##### Stage: 4 Challenge Level:
A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?
### Nine Colours
##### Stage: 3 Challenge Level:
Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
### Archery
##### Stage: 3 Challenge Level:
Imagine picking up a bow and some arrows and attempting to hit the target a few times. Can you work out the settings for the sight that give you the best chance of gaining a high score?
### Teddy Town
##### Stage: 1, 2 and 3 Challenge Level:
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
### Got it Article
##### Stage: 2 and 3
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
### Instant Insanity
##### Stage: 3, 4 and 5 Challenge Level:
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
### Nim-interactive
##### Stage: 3 and 4 Challenge Level:
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.
### Flippin' Discs
##### Stage: 3 Challenge Level:
Identical discs are flipped in the air. You win if all of the faces show the same colour. Can you calculate the probability of winning with n discs?
### Subtended Angles
##### Stage: 3 Challenge Level:
What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
### Speeding Up, Slowing Down
##### Stage: 3 Challenge Level:
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.
### Shuffle Shriek
##### Stage: 3 Challenge Level:
Can you find all the 4-ball shuffles?
### Magic Potting Sheds
##### Stage: 3 Challenge Level:
Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
### Up and Across
##### Stage: 3 Challenge Level:
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its vertical and horizontal movement at each stage.
### An Unhappy End
##### Stage: 3 Challenge Level:
Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...
|
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It was five years ago this month that ESA’s GOCE gravity-mapping satellite finally gave way to gravity, but its results are still yielding buried treasure – giving a new view of the remnants of lost continents hidden deep under the ice sheet of Antarctica.
A research team from Germany’s Kiel University and the British Antarctic Survey published their latest GOCE-based findings this week in the journal Scientific Reports.
Dubbed ‘the Formula one of space’, the GOCE (Gravity field and Ocean Circulation Explorer) mission orbited Earth for more than four years, from March 2009 to November 2013. This sleek, finned satellite with no moving parts was designed around a single goal: to measure the pull of Earth’s gravity more precisely than any mission before.
GOCE flew at an altitude of just 255 km, more than 500 km nearer than a typical Earth observation satellite, to maximise its sensitivity to gravity.
In its last year in orbit, with its supply of xenon propellant holding out well, GOCE was manoeuvred down still lower, to just 225 km altitude, for even more accurate gravity measurements. The propellant keeping it resistant to air drag was finally spent in October 2013, and it reentered the atmosphere three weeks later.
GOCE’s main output was a high-fidelity global gravity map or ‘geoid’, but the mission also charted localised gravity gradients – measurements of how rapidly the acceleration of gravity changes – across all directions of motion, down to a resolution of 80 km.
The team from Kiel University and BAS has converted this patchwork of 3D gravity measurements into curvature-based ‘shape indexes’ across the different regions of our planet, analogous to contours on a map.
The study’s lead author Prof Jörg Ebbing from Kiel University comments, “The satellite gravity data can be combined with seismological data to produce more consistent images of the crust and upper mantle in 3D, which is crucial to understand how plate tectonics and deep mantle dynamics interact.”
In combination with existing seismological data, these gravity gradients show high sensitivity to known features of Earth’s ‘lithosphere’, the solid crust and that section of the molten mantle beneath it.
These features include dense rocky zones called cratons – remnants of ancient continents found at the heart of modern continental plates – highly folded ‘orogen’ regions associated with mountain ranges and the thinner crust of ocean beds.
The new window into the deep subsurface offered by this data offers novel insights into the structure of all Earth’s continents, but especially Antarctica. With more than 98% of its surface covered by ice with an average thickness of 2 km, the southern continent largely remains a blank spot on current geological maps.
“These gravity images are revolutionising our ability to study the least understood continent on Earth, Antarctica,” says co-author Fausto Ferraccioli, Science Leader of Geology and Geophysics at BAS.
“In East Antarctica we see an exciting mosaic of geological features that reveal fundamental similarities and differences between the crust beneath Antarctica and other continents it was joined to until 160 million years ago.”
The gravity gradient findings show West Antarctica has a thinner crust and lithosphere compared to that of East Antarctica, which is made up of a mosaic of old cratons separated by younger orogens, revealing a family likeness to Australia and India.
These findings are of more than purely historic geological interest. They give clues to how Antarctica’s continental structure is influencing the behavior of ice sheets and how rapidly Antarctica regions will rebound in response to melting ice.
ESA’s GOCE mission scientist Roger Haagmans adds, “It is exciting to see that direct use of the gravity gradients, which were measured for the first time ever with GOCE, leads to a fresh independent look inside Earth – even below a thick sheet of ice.
“It also provides context of how continents were possibly connected in the past before they drifted apart owing to plate motion.”
Source: European Space Agency (ESA)
Date: Nov 7, 2018
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## Quantitative Reasoning Questions and Answers Part-2
Data for Questions 1 to 4 : Rajeev planted some plants in his lawn but in certain fixed pattern:
i. In most of the rows there are neither Roses nor Marigolds.
ii. There are two more rows of Orchids than Tulips and two more rows of Roses than Orchids.
iii. There are four more rows of Roses than Tulips.
iv. There aren’t as many rows of Lilly as Fireball.
v. There is one less Marigold row than Rose.
vi. There is just one row of Tulips.
vii. The maximum number of rows he planted is six
1. How many rows of rose did he planted?
a) Two
b) Five
c) Four
d) Cannot be determined
Explanation: From clues 1, 2, 3, 4, 5 we get:
Orchids = Tulips + 2, Rose = Tulips + 4 Marigold = Tulips + 3 and since Lily < Fireball.
If Tulips is 1 (Clue 6), we get
Tulip = 1, Orchids = 3, Marigold = 4, Rose = 5
Hence, Lily = 2 and Fireball = 6
The answer is Five. Option (b) is correct
2. Which of the above information is redundant and can be dispensed with?
a) (i)
b) (iii)
c) (i) and (iii) both
d) All are necessary
Explanation: Statement (iii) is redundant. Option (b) is correct (Refer explanation of Question 1)
3. What is the sum of the rows of Orchids and Marigold he planted?
a) three
b) nine
c) Seven
d) Cannot be determined
Explanation: 3 + 4 = 7. Option (c) is correct (Refer explanation of Question 1)
4. How many rows of fireball did he plant?
a) Two
b) Six
c) Two or Six
Explanation: Six. Option (b) is correct (Refer explanation of Question 1)
Data for Questions 5 to 10 : In a class of 540 students, for every 9 girls these are 11 boys. The weight of students varies from 40 to 50 kg. There are as many 44 kg girls as there are 46 kg boys and as many 40 kg boys as 50 kg girls. The number of 50 kg boys is 35 more than that of 44 kg girls while there are as many 44 kg boys as 46 kg girls. The ratio of 40 kg boys and girls is 4:3 while that of 50 kg girls and boys is 1:3
5.How many boys weigh 40 kg?
a) 22
b) 24
c) 28
d) None of these
Explanation:
Start from the fourth line and take ‘a’ as the number of girls of 44 kgs.
The answer is 24. Option (b) is correct
6. How many girls weigh 44 kg?
a) 37
b) 36
c) 39
d) None of these
Explanation: The answer is 37. Option (a) is correct (Refer explanation of Question 5)
7. How many girls weight 46 kg?
a) 165
b) 164
c) 146
d) None of these
Explanation: The answer is 164. Option (b) is correct (Refer explanation of Question 5)
8. The number of boys weighing 50 kg is:
a) 72
b) 74
c) 76
d) None of these
Explanation: The answer is 72. Option (a) is correct (Refer explanation of Question 5)
9. The number of girls weighing 40 kg is:
a) 16
b) 18
c) 22
d) None of these
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# Variables used in Statistics and their types
When running a statistical analysis, it is very likely that you will stumble across terms like ‘ordinal variable’ or ‘nominal variable’. While many of us have a basic idea of what the different types of variables are, the nomenclature can get confusing.
Before embarking on your research journey, selecting your statistical tools and collecting your data, let’s take a quick look at the different types of variables you may encounter. When using different statistical analyses software such as SPSS, you will see these words and when you have a grasp of them makes the learning process much smoother.
• ### Constant:
While this is not a variable, it is still important to understand the difference between a constant and a variable. A constant, to explain it simply, is a measure whose value remains constant. For example, the number of days in a week is constant.
• ### Variable:
In contrast, a measure whose value is not fixed is known as a variable. Age, gender, height and weight are all common examples of variables. The three main types of variables that will be discussed here include categorical, ordinal and interval variables. To better understand these terms, let us refer to the hypothetical example below.
#### Example:
A 5th-grade classroom consists of 10 students. As part of a math class on statistical data collection and frequency analyses, the classroom teacher records the students’ gender, heights and marks in mathematics. The information is as shown below.
S. No. Gender Height Marks 1. Male Tall 70-80 2. Male Tall 80-90 3. Female Medium 70-80 4. Male Tall 90-100 5. Female Short 90-100 6. Female Tall 70-80 7. Male Medium 70-80 8. Female Tall 70-80 9. Female Tall 80-90 10. Male Medium 80-90
In this example, gender is a categorical variable. Categorical variables or nominal variables are those possessing more than one category, but no specific order. For instance, gender, in this case, exists as two groups of either male or female, but there is no arbitrary order for the use of these groupings.
Similar to a categorical variable is an ordinal variable, in this example represented by height. Ordinal variables are in which the categories have some specific order. We know that the order to be considered here is short, medium and tall and it can be assumed that the teacher has defined each category by some fixed constants.
The last type of variable is an interval variable, which is similar to an ordinal variable except that the intervals are equal. The last column lists the students’ marks in mathematics, grouped according to intervals of 10. We can order the categories according to their numerical value and the interval between each category is equal.
### Are there any other types of variables you should know about?
When working with statistical software and tools, you may come across the terms ‘independent’ and ‘dependent’ variables.
An example of this is if we are examining a collection of data about the change in consumption of chocolate according to age.
In this case, age is the independent variable, it does not depend on anything. The change in chocolate consumption becomes the dependent variable as it depends on age.
If you are interested in getting our help for statistical analysis, Click here.
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# Sum of Perfect Squares Formula
Before knowing what is the sum of the perfect squares formula, first, let us recall what are perfect squares. A perfect square is a number that can be written as the square of a number. 1, 4, 9, 16, 25, 36, etc are some perfect squares as they can be expressed as 12, 22, 32, 42, 52, 62, etc respectively. The sum of perfect squares formula is used to find the sum of two or more perfect squares without adding them manually.
## What Is the Sum of Perfect Squares Formula?
We have two types of formulas for finding the sum of perfect squares. One is the formula to find the sum of two perfect squares, the other formula is to find the sum of the first "n" perfect squares.
• ### The formula for finding the sum of the squares for first "n" natural numbers is:12 + 22 + 32 + ... + n2 = [ n (n + 1) (2n + 6) ] / 6
Let us see the applications of the sum of perfect squares formulas in the following section.
## Solved Examples Using Sum of Perfect Squares Formula
### Example 1: Find the sum of squares of 101 and 99.
Solution:
To find: The sum of squares of 101 and 99. i.e., 1012 + 992.
Using the sum of perfect squares formula:
a2 + b2 = (a + b)2 - 2ab
Substitute a = 101 and b = 99 in the above formula:
1012 + 992 = (101 + 99)2 - 2(101)(99)
= (200)2 - 2 (9999)
= 40000 - 19998
= 20,002
### Example 2: Find the sum of squares of the first 25 natural numbers.
Solution:
To find: The sum of the first 25 natural numbers.
Substitute n = 25 in the sum of the perfect squares formula of the first n natural numbers:
12 + 22 + 32 + ... + n2 = [ n (n + 1) (2n + 6) ] / 6
12 + 22 + 32 + ... + 252 = [ 25 (25 + 1) (2(25) + 1) ] / 6
= ( 25 × 26 × 51 ) / 6
= 5,525
Answer: The sum of the first 25 natural numbers = 5,525.
|
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|Simon Says||Snack #2|
|Description:||Students mimic healthy snack foods (pop like popcorn, wiggle like string cheese); but if Simon doesnít say it, watch out!|
|Objective:||Students will recognize the importance of eating a variety of foods for snack.|
- Have the students line up against the wall at one end of the room facing you.
- Quickly review why it is important to eat different kinds, or a variety, of healthy foods every day (because each one does something different and special for our bodies).
- Give a few examples of ways different healthy snack foods help us: Carrots help our eyesight, popcorn helps us digest (poop), low-fat or skim milk helps our bones grow strong.
- Tell them you are going to think about the many healthy snack foods they can choose from while you play "Simon Says."
- Explain that you (as Simon) are the only person the students should listen to. Tell them you will command them to do certain movements. If you donít say "Simon says" before the command, the class should not do the action.
- Say the following (see below for more):
- "Simon Says: Pop like popcorn."
Encourage them to hop up and down.
- "Simon Says: Grow tall towards the sky like an apple tree."
Encourage them to reach up high.
- "Wiggle like string cheese."
- "Simon Says: Go nuts like nuts."
Encourage them to dance around waving their arms.
- "Be round like an orange."
- "Flow like water."
- "Simon says: Be stiff like carrot sticks."
Encourage them to stand still with their arms at their sides.
- "Simon Says: Wrinkle like a raisin."
Encourage them to scrunch up and bend over.
Children should be encouraged to consume a variety of nutrient-rich foods low in fat and added sugar for snack. There are five food groupings:
- milk and milk products
- meats, beans, nuts
Children should eat foods from all five groupings everyday.
More Simon Says Actions:
- "Simon Says: Form a bunch like grapes."
Encourage students to get into small groups and hold hands.
- "Simon Says: Buzz like a honey bee."
Encourage students to make buzzing sounds as they jog around the room.
- "Simon Says: Twist in a knot like a pretzel.
Encourage students to twist their legs and wrap their arms around their bodies.
- "Simon Says: Bend like a banana."
Encourage students to curve their spines as they bend over.
|
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In the basement of the University of Arizona’s Laboratory of Tree-Ring Research, the fragrant smell of pine hangs in the air as researchers comb through the stacks of tree slabs to find a round, 2-inch-thick piece of Douglas fir.
They point out an anomaly in the slab — an unusually wide set of rings that represent the years 1905 to 1922. Those rings mean it was a pluvial period — precipitation was well above average — and so the trees grew far more than other years.
“In 1905, the gates opened and it was very wet and stayed very wet until the 1920s,” said David Meko, a hydrologist at the lab who studies past climate and stream flow based on tree rings. “It guided their planning and how much water they thought was available.”
The planning was that of the states that share the water of the Colorado River. Worried that a burgeoning California would take most of the water before it was fairly divvied up, representatives from the other Colorado River Basin states, presided over by U.S. Secretary of Commerce Herbert Hoover, came together in 1922 to develop an equitable apportionment. They looked at flow measurements and figured that the river contained an average of 15 million acre-feet. They divided the Colorado River states into two divisions – the upper basin and the lower basin, with the dividing line in northern Arizona near the Utah border. The upper basin states — Utah, Wyoming, Colorado, and New Mexico — agreed not to take more than a total of 7.5 million acre-feet and to allow the other half to flow south to the lower basin. The agreement they signed was called the 1922 Colorado River Compact, also known as the Law of the River.
The 1922 compact, though, is based on a premise that the tree rings in the University of Arizona lab now show is false. The river’s long-term average flow is about 12 to 15 million acre-feet, in a good year. Meanwhile, the lower basin states — Arizona, California, and Nevada — use 7.5 million acre-feet, and in 1922 no one factored in evaporative losses from the desert sun at the yet-unbuilt Lake Mead reservoir, which amount to another 1.2 million acre-feet, or the water taken up by plants. Nor did anyone factor in a subsequent 1944 treaty that requires the United States to provide 1.5 million acre-feet to Mexico. A conservative estimate on how much Colorado River water is actually used is 20 million acre-feet.
This over-appropriation is problem enough, but in recent years the river’s flow has been dwindling. The region is locked in a 19-year-long drought, the most severe in 1,250 years. And it may continue much longer. The tree ring data shows that there have been numerous multi-decadal or mega-droughts in the basin in the last 1,000 years. The prospect that drought could be the new normal for the region is creating a good deal of anxiety along the Colorado.
“Many water managers like me are struggling at not panicking,” said Mark Harris, general manager of the Grand Valley Water User’s Association in Grand Junction, Colorado. In his farm cap and jeans, Harris is a no-nonsense type, not given to hyperbole. This year, though, some “junior” water users on the Yampa River, a tributary to the Colorado, were told they would not get their water because others had priority, the first time that has ever happened, and late-season water flows near Grand Junction were near crisis levels. “The crunch is here,” Harris said. “It’s here, and it will stay here. We will never be out of the woods, we are in the woods forever.”
Another low-snow winter would trigger the first emergency declaration in the basin, forcing states to deal with water cutbacks.
Never has the question of “what will the winter be like?” loomed larger than it does this year in the Colorado River Basin. If it is anything like last year (when about two-thirds of the usual snow fell) and many other low snow years since 2000, it will trigger the first emergency declaration in the basin, which could force states to deal with cutbacks in the water they are appropriated. And even if it is a big snow year, it will likely only delay what now seems inevitable.
The last time Lake Mead was full was 1983. Since then it has slowly declined. It is now 40 percent full: 1,082 feet above sea level. It may never be full again, experts say. If it drops 7 feet, to 1,075 feet, it will trigger the first Tier 1 water cutbacks. A flyover reveals a giant white ring all the way around the lake’s 112-mile-long perimeter, dramatically showing how far water levels have dropped.
There are three levels of cutbacks. When Lake Mead falls to 1,050 feet, a Tier 2 crisis occurs, and Tier 3 at 1,025. At each level, states in the lower basin have to give up more of their water. Lake Mead would have already hit 1,075 feet and a First Tier declaration if it weren’t for the fact that farmers, ranchers, and many others have been working to avoid an emergency by keeping more water in the river through conservation efforts. For example, in 2017, state, federal, municipal, and private entities funded the purchase of 40,000 acre-feet from the Gila River Indian Community to be left in Lake Mead in perpetuity as part of a system conservation agreement.
Last August, the U.S. Bureau of Reclamation issued a report on the water future of the region. The agency’s predictions were sobering. By May of this year, the bureau forecast the level will dip just below 1,075 feet, and at the beginning of 2020, the level is expected to drop to 1,070. By the summer of 2020, the prediction is 1,050 feet, almost Tier 2. If these predictions come true, users will have to begin giving up their water allotments, starting with the most junior.
If water levels continue to drop, sinking below 1,050 feet, Hoover Dam — which impounds Lake Mead and provides power to millions of people in Southern California, Nevada, and Arizona — will stop generating electricity, as water levels will be too low to flow through it. And should Lake Mead keep dropping all the way to 895 feet, it will fall below the level at which water can be piped out — the dreaded “dead pool.” Moreover, because Lake Mead is funnel-shaped, the lower it gets the faster it drops. At some point there is the likelihood that the lower basin will force the upper basin to send water to meet its obligations — a compact call — something that’s never happened before.
A few wet years in a long dry spell would be critical these days to keep the Colorado from completely drying up.
All of this is uncharted crisis terrain. “If the drought is multi-decadal the system will fail,” said Jack Schmidt, a professor of watershed science at Utah State University. “But nobody knows what failure means.”
Arizona officials have a sense of it and are coming to grips with the reality. They are the most junior users in the Lower Basin and a Tier 1 shortage would mean Arizona would have to start cutting allocations to users. “If the current climate trend continues,” said Kathryn Sorensen, director of the Phoenix Water Services Department, “you could have ‘dead pool’ in four years. That’s worst case.” Should that happen, the whole region, she says, would be thrown into crisis.
If these were normal times, past droughts might give us a sense of what might be in store. The climate information stored in tree rings show that the longest drought in this region occurred in medieval times and lasted for 62 years — with no very wet years in between the dry ones. A few very wet years in a long dry spell would be critical on the Colorado these days to keep it from completely drying up.
But it may be even worse than that. This drought is unusually hot. “Temperatures keep going up,” said Meko, of the University of Arizona tree ring lab. “We keep breaking records year after year. It’s additional stress on the water system.” Meanwhile, the two driest years all the way back to the 1200s occurred in 1996 and 2002. “It’s a little worrisome to see the most extreme years right near the present,” he said.
“Droughts impacted by warmer temperatures will be more severe,” says Connie Woodhouse, who also studies climate at the tree ring lab. “A lack of precipitation is one thing. But when a drought happens and temperatures are warmer, the precipitation deficits are exacerbated. You have more evaporation, more ground heating, and it impacts the snowpack.”
From 2000 to 2014, flows in the river were 19 percent below the averages in the previous years, and a third of that loss was caused by high temperatures, according to researchers Jonathan Overpeck of the University of Michigan and and Brad Udall at the Colorado Water Insitute at Colorado State University, in an often-cited paper about the unprecedented nature of this drought and what it means for the future.
The biggest impact of high temperatures is something called runoff efficiency — the amount of stream flow that results from precipitation. Right now about 15 percent of the water in the snow in the watershed makes it into the river. The other 85 percent soaks into the ground, evaporates, or is taken up by plants. As it gets warmer, runoff efficiency is decreasing. Shorter winters mean the ground has less snow cover and is darker, so it warms up more and sooner, which means snow melts faster and more water evaporates and is taken up by plants in a longer growing season. The Colorado River Research Group, 10 veteran academics who study the Colorado, call this most alarming change to the physical environment.
The alarm is palpable among water managers throughout the Southwest. They see the writing on the wall.
Warmer temperatures also mean that of the precipation that does come, more of it will fall as rain instead of snow. The Colorado’s engineering infrastructure was built around the natural long-term storage that snowpack provides, but rain pulses quickly through the system.
Meanwhile, the rapid development of everything from housing developments to solar installations in the Southwest has created more dust particles which go airborne and settle on to the snow fields of the Rockies, five to seven times as much dust as was seen a century ago. The darker snow melts sooner and faster, a phenomenon that costs the river about 5 percent of its flow. And as the drought continues, there’s more dust from more dry ground and that creates more dust.
As the flow of the Colorado diminishes, more water users will be forced to turn to groundwater pumping.The news on that front, though, is also problematic. In a 2014 paper, researchers at the Global Institute for Water Security, which uses a satellite to measure large-scale changes in groundwater by measuring changes in gravitational pull, found that from 2004 to 2013, the loss of groundwater from pumping was 6.5 times greater than the total loss of water from Lake Powell and Lake Mead. “Everybody knows that groundwater will become progressively more important,” said Jay Famiglietti, the institute’s director. “The problem is groundwater is rapidly disappearing so we shouldn’t depend on it being there.”
However the biggest cloud looming over the Colorado River Basin is whether the region is entering a completely new era, a permanent change as opposed to a temporary one, caused by a planet being rapidly warmed by human activity. “Is this a drought or is this aridification of the Southwest and Colorado River Basin?” asked the University of Michigan’s Overpeck, who has long studied the Colorado.
Like Overpeck, many experts believe the drying up of the Colorado is being driven by a changing climate. “It’s going to get drier and drier,” he said. “It could mean a hell of a lot less water in the river. We’ve seen declines of 20 percent, but it could get up to 50 percent or worse later in this century.”
If climate change is locked in, he said, what is going on now is not a new normal, but a stop along the way to a yet-drier new normal somewhere in the distant future. “In that case, every year will be a new reality,” he said. “The aridification of the Southwest will continue as long as we put greenhouse gases into the atmosphere. We need to stop burning fossil fuels and that will help stop the decline in the river flow.
“And even if we did that, there’s warming baked in,” he said. “It would continue for another decade and then stabilize. Then we will get the new normal. And it will be at that level of warmth for centuries.”
That’s why the alarm is palpable among water managers in the Southwest. They see the writing on the wall, and there are few skeptics about climate change among them. The plight of Cape Town, South Africa, which came to the brink of a water system crash last year, is on many people’s minds along the Colorado River.
This era of drying is especially serious because so much — some 40 million people and an economy that includes the world’s fifth largest, in California — is riding on the flow of the Colorado. The specter of a region facing an existential crisis because of a warming climate becomes more real every day. “If you can see it, you should plan for it,” Phoenix’s Sorensen said. “And I can see it.”
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Friday, December 1, 2023
# PARCC Algebra 2: Modeling a delivery route
-
The following constructed-response question, explained here in hopes of helping algebra 2 students and their parents in Maryland prepare for the PARCC test near the end of this school year, appears on the released version of the PARCC Algebra 2 sample items released following the 2015 test.
The arrangement of a distribution center and four stores to which it delivers is shown on the grid. Each unit on the grid represents 5 miles. The grid lines represent the roads.
The distribution center operators will use a single vehicle and must decide between a large truck and a small van. They will base their decision on this information.
Large Truck
• fuel efficiency: 9 miles per gallon
• delivers to all stores in one round-trip
• uses the shortest route to go to Stores A through D and then back to the distribution center
We can, of course, determine the exact point at which it will become more expensive to use the small van and skip the $12 fixed fees but buy more gas. The two graphs intersect at the point where $8\frac{8}{9}x = 6\frac{2}{3} x + 12$ Remember, in our model, x is the price of gas in dollars per gallon. $8\frac{8}{9}x - 6\frac{6}{9}x = 12$ $2\frac{2}{9}x = 12$ $x = 5\frac{2}{5} = 5.40$ So, a little less than my eyeball estimate of$5.50, so I conclude my answer makes sense. If gas is more expensive than that, it will be better for the company to use the large truck to make the deliveries and pay the extra $12 fee for docking, because the truck won’t have to return to the distribution center each time and guzzle all that extra gas. ## Rounding differences Note that the price for gas I found in part B differs from that shown on the PARCC scoring guidelines for this problem. Instead of keeping exact numbers, PARCC rounded to the nearest tenth in the models used for part A. That gave them an answer of$5.45 in part B, which is just as correct as mine, given that PARCC rounded.
They did all the right computations in their work and justification and executed them without error. Rounding is not an error, but it resulted in a difference in the exact number of 5 cents.
A lot was being made a few years ago about how “Common Core math” seemed to say several different answers would be considered correct, as long as they were justified properly, given the correct mathematics. That didn’t sit very well with some parents, who also thought everything we teach third graders through high school students has to meet the exacting standards of a graduate engineering student. This problem is a prime example of where an answer of $5.45 can be just as acceptable, and earn the student just as many points, as$5.40.
On a multiple-choice test, the two different answers would never be available to students, and on a constructed response question like this one on the PARCC test, the student gets most of the points on this problem for the modeling and the justification, not for the correct numerical value for the price of gas.
Paul Katulahttps://news.schoolsdo.org
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.
### Star Sportsmanship on a Wis. Cross-Country Track
0
Two cross-country runners displayed great sportsmanship at a Wis. meet as they stopped short of the finish line to help a competitor.
|
crawl-data/CC-MAIN-2023-50/segments/1700679100452.79/warc/CC-MAIN-20231202203800-20231202233800-00578.warc.gz
| null |
# Binomial Distribution in R
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It’s a fundamental distribution in statistics and finds applications in various fields such as finance, insurance, quality control, and social sciences.
In this comprehensive article, we’ll explore the binomial distribution, how to generate and work with binomial distributions in R, the functions associated with binomial distributions, and practical applications of the binomial distribution in R.
## Understanding the Binomial Distribution
A binomial distribution is characterized by three parameters:
1. The number of trials (n): This is the fixed number of Bernoulli trials. A Bernoulli trial is an experiment where the outcome can be classified as either a success or a failure.
2. The probability of success in each trial (p): This is the probability that each Bernoulli trial will result in a success.
3. The number of successes (k): This is the number of successes we are interested in when conducting n Bernoulli trials.
The probability mass function of the binomial distribution is given by:
P(X = k) = C(n, k) * (p^k) * (1 - p)^(n - k)
where C(n, k) represents the number of combinations of n items taken k at a time, and p is the probability of success on each trial. The mean and variance of a binomial distribution are np and np(1 - p), respectively.
## Binomial Distribution Functions in R
R provides four functions to work with the binomial distribution:
1. dbinom(x, size, prob, log = FALSE): The density function. This gives the probability of getting x successes in size trials. If log = TRUE, it returns the log-probabilities.
2. pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE): The distribution function. This gives the cumulative probability of getting q or fewer successes. If lower.tail = FALSE, it returns the survival function 1 - pbinom(q). If log.p = TRUE, it gives the log-cumulative probabilities.
3. qbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE): The quantile function. This gives the number of successes corresponding to a certain cumulative probability p.
4. rbinom(n, size, prob): This generates n random numbers from a binomial distribution.
## Generating a Binomial Distribution in R
You can generate a binomial distribution in R using the rbinom() function. Here’s an example:
set.seed(123) # for reproducibility
x <- rbinom(1000, size = 10, prob = 0.5)
This code generates a dataset x of 1000 observations drawn from a binomial distribution with 10 trials and a probability of success of 0.5.
## Visualizing a Binomial Distribution in R
You can visualize a binomial distribution using a histogram or a bar plot. For a binomial distribution, a bar plot may be more appropriate because it’s a discrete distribution. Here’s an example:
barplot(table(x)/length(x),
main = "Binomial Distribution",
xlab = "Number of successes",
ylab = "Probability")
## Computing Probability and Quantiles
You can calculate the probability of obtaining a certain number of successes using the dbinom() function. Similarly, pbinom() and qbinom() can be used to find the cumulative probability and the number of successes for a certain percentile (quantile), respectively.
Here’s an example:
# Probability of getting exactly 5 successes
prob <- dbinom(5, size = 10, prob = 0.5)
print(prob)
# Cumulative probability of getting 5 or fewer successes
cum_prob <- pbinom(5, size = 10, prob = 0.5)
print(cum_prob)
# Number of successes at the 90th percentile
quantile <- qbinom(0.90, size = 10, prob = 0.5)
print(quantile)
## Applications of Binomial Distribution in R
The binomial distribution has numerous applications in R:
1. Survey Analysis: If you are conducting a survey and want to know the probability of a certain number of people responding positively, you can use the binomial distribution to model the outcomes.
2. Quality Control: The binomial distribution can be used to model the number of defective items in a batch of products.
3. Risk Assessment: In insurance or finance, the binomial distribution can be used to model the number of claims or defaults.
4. AB Testing: In online experiments, the binomial distribution can be used to model the number of successes (clicks, conversions, etc.) out of a number of trials (website visits, emails sent, etc.).
## Conclusion
The binomial distribution is a fundamental distribution in statistics that describes the probability of obtaining a certain number of successes in a fixed number of Bernoulli trials. Understanding the binomial distribution and how to work with it in R is a vital skill for anyone involved in statistical analysis or data science. R provides robust capabilities to work with binomial distributions, making it a powerful tool for statistical modeling and hypothesis testing.
Posted in RTagged
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| null |
Spinal cord injury
Every year, about 2000 people in the UK, and several million people worldwide, suffer traumatic spinal cord injury leading to permanent paralysis. Average age at injury is 31, with the greatest frequency between 15 and 25 years. About four times as many have spinal cord lesions at birth or caused by disease. However, according to a 2002 review article in The Lancet
The initial trauma includes both traction, which pulls nerve cells apart, and compression, which damages nerves and blood vessels. Nerve fibres that are detached from their cell nucleus must be rejoined within 48–72 hours or function is lost forever. The spinal cord swells within minutes, and there is further loss of blood supply when the pressure in the spinal canal rises. Lack of blood to the injured tissues, chemicals from disrupted nerve membranes, and electrolyte shifts trigger a cascade of secondary injuries that harm or kill neighbouring cells. Finally, scar tissues fills the void.
Most basic research in spinal injury is done on rats. However spinal injury is common in dogs, particularly dachshunds, and many dog owners are glad to let research be performed on their pets, which would otherwise have to be put down. Researchers at Cambridge University in the UK , and at Purdue University in the USA , in collaboration with local vets, are treating pet dogs with increasing success. Human clinical trials arising from dog results started at Purdue in 2004.
Researchers have discovered that rats with incomplete severance of the spinal cord have some capacity for regrowing fibres. When 3% of the forepaw connections remain, rats gradually recovered co-ordinated movements. Study of their spinal cords under the microscope showed spontaneous sprouting from nerve fibres spared by the original injury. This explains why some spontaneous motor recovery takes place after partial (40%) spinal cord injury, stroke and head trauma. The researchers are now testing whether sprouting can be promoted experimentally, by delivering nerve growth factors - proteins that stimulate nerve growth - to the injury site
In 2003, scientists performed a meticulous series of experiments with rats, showing for the first time the presence of a signalling system called Wnt/frazzled that directs regrowing neurons to the right place after spinal injury .
Other naturally-occurring chemicals can help regenerating nerves to re-enter the spinal cord . In embryos, the signalling protein named WNT-3 directs specific motor nerve cells to the correct connections in the spine, and this might be harnessed to aid recovery in injured adult animals .
An attractively simple approach to treating partial spinal injury is to inject a common antibiotic, minocycline, within one hour of injury. It reduces tissue loss by blocking release of a protein called cytochrome c .
An important factor that inhibits recovery is the scarring that takes place after injury. When a barrier stopped inflammatory cells from reaching the injured site, nerve cells on both sides of the injury site were able to grow and re-establish connections with each other over two to three weeks, leading to substantial recovery of function. A compound that stops the growth of new blood vessels has also enabled mice with spinal cord injuries to walk again , presumably by reducing scarring. A naturally occurring anti-inflammatory and anti-scarring agent, decorin, allowed nerve fibres to grow across nerve injuries in four days .
Implanting specially treated immune system cells has also been successful
The immune system's involvement is complex: scientists have now shown in both rats and mice that central nervous system damage triggers an autoimmune reaction that actually protects nerve cells from further damage. This finding may lead to a vaccine to improve recovery following spinal cord injury and inhibit the cascade of damage that occurs after the initial trauma . Rats vaccinated with protein fragments from the central nervous system soon after partial injury to the spinal cord showed significant recovery of movement and more healthy nerve fibres in the spinal cord than untreated rats .
Nerve transplants, usually from rat fetuses, have been shown to bridge spinal cord gaps in adult rats
Implants of embryonic stem cells, which have the ability to develop into any cell type in the body, might be successful. Nerve stem cells have been shown in mice to develop into all types of functioning nerve cell . When mouse stem cells are transplanted into the damaged spinal cords of rats, the rats were able to walk again .
Human stem cells injected into the spinal cord of paralysed rats became astrocyes (support cells) and sensory cells, but about four cells per rat became motor nerve cells that controlled movement, and had the unexpected additional effect of helping the rat nerve cells regrow .
In 2004, scientists reported that they had isolated adult nerve cells and had cultured them two years in a test tube (the longest anyone has kept these self-renewing cells) and injected them into damaged spinal cord of rats, where they replaced missing cells. A reassuring finding was the lack of tumour growth .
In 2003 scientists discovered that brain stem cells, unlike adult brain cells, are immune privileged, which means they can be transplanted into any part of the body without being rejected . In 2004 in a highly-sophisticated experiment, nerve stem cells were started on their development into motor neurons in a test tube and implanted into the spinal cords of injured rats, which were then treated with chemicals that prevented the nerve sheath cells from blocking development. Each rat received about 12,000 neurons, of which about 80 became fully-fledged motor nerve cells that were electrically linked with the spinal cord .
Scientists have achieved substantial success using a combination of nerve growth factor and antibodies that neutralise nerve growth inhibitors
When rib nerves were grafted into the spinal cords of spinal-injured rats, accompanied with a growth factor called aFGF and physical stimulation, hind leg movement was partially restored .
Yet more work is required, but, as The Lancet pointed out, these promising results in rodents hold out the hope that we may soon be able to help human victims of spinal cord damage overcome their paralysis.
- McDonald JW, Sadowsky C (2002) Spinal cord injury. Lancet 359, 417.
- RDS News, January 2001, p.10.
- See http://www.vet.purdue.edu/cpr/
- Weidner N, Ner A, Salimi N & Tuszynski MH (2001) Spontaneous corticospinal axonal plasticity and functional recovery after adult central nervous system injury Proc Nat Acad Sci 98, 3513.
- Lyuksyutova AL, Lu C-C, Milanesio N (2003) Anterior-posterior guidance of commissural axons by Wnt-Frizzled signaling. Science 302, 1984.
- Ramer MS, Priestley JV, & McMahon SB (2000) Functional regeneration of sensory axons into the adult spinal cord Nature 403, 312.
- Grandpré T, Li S & Strittmatter (2002) Nogo-66 receptor antagonist peptide promotes axonal regeneration Nature Neuroscience 417, 547.
- Neumann S, Bradke F, Tessier-Lavigne M & Basbaum A (2002) Regeneration of sensory axons within the injured spinal cord induced by intraganglionic camp elevation Neuron 34, 885.
- Qui J, Cai, D, Dai H et al (2002) Spinal axon regeneration induced by elevation of cyclic AMP Neuron 34, 895.
- Krylova O, Herreros J, Cleverley KE et al (2002) WNT-3, expressed by motoneurons, regulates terminal arborization of neurotrophin-3-responsive spinal sensory neurons Neuron 35, 1043.
- Teng YD, Choi H, Onario RC et al (2004) Minocycline inhibits contusion-triggered mitochondrial cytochrome c release and mitigates functional deficits after spinal cord injury. Proc Nat Acad Sci 101, 3071.
- Seitz A, Aglow E, Heber-Katz E (2002) Recovery from spinal cord injury: A new transection model in the C57Bl/6 mouse J Neurosci Res 67, 337.
- Wamil AW, Wamil BD, Hellerqvist CG (1998) CM101-mediated recovery of walking ability in adult mice paralyzed by spinal cord injury Proc Nat Acad Sci 95, 13188.
- Davies JE, Tang X, Denning JW et al (2004) Decorin suppresses neurocan, brevican, phosphacan and NG2 expression and promotes axon growth across adult rat spinal cord injuries. Eur J Neurosci 19, 1226.
- Rapalino O, Lazarov-Spiegler O, Agranov E et al (1998) Implantation of stimulated homologous macrophages results in partial recovery of paraplegic rats. Nature Medicine 4, 814.
- Neumann S & Woolf CJ (1999) Regeneration of dorsal column fibers into and beyond the lesion site following adult spinal cord injury Neuron 22, 83.
- Yoles E, Hauben E, Palgi O et al (2001) Protective autoimmunity is a physiological response to CNS trauma J Neurosci 21, 3740.
- Hauben E, Agranov E, Gothilf A et al (2001) Post traumatic therapeutic vaccination with modified myelin self-antigen prevents complete paralysis while avoiding autoimmune disease J Clin Invest 108, 108.
- Iwashita Y Kawaguchi S & Murata M (1994) Restoration of function of spinal cord segments in the rat Nature 367, 167.
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- Davies SJA, Fitch MT, Memberg SP et al (1997) Regeneration of adult axons in white matter tracts of the central nervous system Nature 390, 680.
- Imaizumi T, Lankford KL, Burton WV et al (2001) Xenotransplantation of transgenic pig olfactory ensheathing cells promotes axonal egeneration in rat spinal cord Nature Biotech, 18, 949.
- Saporta S, Makoui AS, Willing A et al (2002) Functional recovery after complete contusion injury to the spinal cord and transplantation of human neuroteratocarcinoma neurons in rats J Neurosurg: Spine 97, 63.
- Silva GA, Czeisler C, Niece KL et al (2004) Selective differentiation of neural progenitor cells by high-epitope density nanofibers. Science 303, 1352.
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Last edited: 25 September 2018 15:49
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# calculate E=(a^(-1)+b^(-1))/a^(-1)-b^(-1) if: a=((5√3+√50)(5-√24))/√75-5√2 b=√(7+4√3)+√(7-4√3)
sciencesolve | Teacher | (Level 3) Educator Emeritus
Posted on
You need to evaluate `a^(-1)` and `b^(-1)` such that:
`a^(-1) = 1/a, b^(-1) = 1/b`
`a^(-1) = (sqrt 75 - 5sqrt 2)/((5sqrt 3 + sqrt 50)(5 - sqrt 24))`
`a^(-1) = (sqrt 75 - 5sqrt 2)/(25sqrt 3 - 30sqrt 2 + 25sqrt 2 - 20sqrt3)`
`a^(-1) = (5sqrt 3 - 5sqrt 2)/(5sqrt 3 - 5sqrt 2)`
Reducing duplicate factors yields:
`a^(-1) = 1`
You need to evaluate `b^(-1)` such that:
`b^(-1) = 1/(sqrt(7+4sqrt3) + sqrt(7-4sqrt3))`
`b^(-1) = (sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(7+4sqrt3-7+4sqrt3)`
`b^(-1) = (sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(8sqrt 3)`
Evaluating E yields:
`E = (1 + (sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(8sqrt 3))/((sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(8sqrt 3))`
Reducing duplicate factors yields:
`E = (8sqrt 3 + sqrt(7+4sqrt3) - sqrt(7-4sqrt3))/(sqrt(7+4sqrt3) - sqrt(7-4sqrt3))`
`E = ((8sqrt 3 + sqrt(7+4sqrt3) - sqrt(7-4sqrt3))(sqrt(7+4sqrt3) + sqrt(7-4sqrt3)))/(8sqrt 3)`
Hence, evaluating the given expression yields `E = ((8sqrt 3 + sqrt(7+4sqrt3) - sqrt(7-4sqrt3))(sqrt(7+4sqrt3) + sqrt(7-4sqrt3)))/(8sqrt 3).`
oldnick | (Level 1) Valedictorian
Posted on
First wirte `(a^(-1)+b^(-1))/(a^(-1)-b^(-1))` as `(1/a+1/b)/(1/a-1/b)=` `((b+a)/(ab))/((b-a)/(ab))` `=(b+a)/(b-a)`
Now `a=((sqrt(3)+sqrt(2))(5-2sqrt(6)))/(sqrt(3)-sqrt(2))`
multypling for `(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))` we get:
`a=(5+2sqrt(6))(5-2sqrt(6))=1`
`b= sqrt(7+4sqrt(3))+sqrt(7-4sqrt(3))`
`b= sqrt(7+sqrt(48))+sqrt(7-sqrt(48))`
`b=2sqrt((7+sqrt(49-48))/2)` `=+-4`
So E has two values:
`E_1=(4+1)/(4-1)=5/3`
`E_2=(-4+1)/(-4-1)=3/5`
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The social model of disability states that disability is not caused by a persons impairment or difference but by how society is structured and organized. Disability is not only within body or mind of a person but in the interaction between individuals with intellectual, bodily differences and their social environment. The model looks at ways to remove these barriers and negative attitudes that limit life choices of disabled persons. If these obstacles are removed, disabled persons become equal and independent in the society. They, therefore, have control and choices in their lives. This model was developed to replace the traditional medical model that was used to diagnose and treat disabled persons. This model advocates for ameliorating disability via a change in culture, social policies and institutional practices. An essential facet of this model concerns equality. It seeks to change society to tolerate persons with disability. This model focuses on changes required in society regarding attitudes, a positive attitude by valuing those impaired. Social support by removing barriers, availing resources and aids. Using suitable formats to convey information. Enhancing physical structures for example elevators. The social model differs with the medical model of disability. Assistive technology refers to assistive tools that are used by people with disabilities for rehabilitative and adaptive purposes. They include equipment, product systems that are used to increase and upgrade functional abilities of disabled persons. These devices enable independence by allowing disabled people to perform tasks that they were initially not able to do or were difficult (Shinohara & Wobbrock 2016). In this study am going to discuss how an understanding of the social model of disability and the availability of assistive technology help care practitioners create an enabling environment:
People with physical impairments that affect locomotion can use mobility aid devices, for example, wheelchairs, walkers crutches, and even canes. Wheelchairs are utilized by individuals whose walking is hard or impossible in cases of injury, illness or disability. This technology under the social model solution requires a ramp at the entrance to enable individuals to get into a building with ease. Walkers and crutches are used to maintain balance and stability.
Persons with visual impairments may need to reside at a public place to get help when they need it. These people also need to use assistive technology like screen readers, large print books, voice recording devices, magnifiers, and Braille. Screen readers enable easy access to electronic information. They convert text into audio. Braille consists of raised dots that represents information and can be read using fingers. Desktop magnifiers for video are devices that perform magnification to the printed materials. Large print keyboards have big printed letters on the keys.
People with hearing disability require an electroacoustic device that amplifies sound. They also depend on visual and tactile channels to receive and convey information. The device helps individuals fully participate in community activities by hearing more. Other assistive devices that could enhance hearing include digital in-ear, behind the ear and on the body aids, captions on TV and keying handsets. Additionally, people with cognitive impairments require assistive tools for memory aids, for example, the smart pen which transforms written notes into digital and audio recordings .computers and other electrical devices are used to help people following an injury to the brain (Carver et al. 2016).
Educational software is used to assist individuals with learning, reading, and comprehensive disabilities. These include word predictions note takers text enlargers and book readers. Other devices page turners, pencil grips, and book holders enable learners with impairments to fully participate in learning activities. Assistive technology devices have allowed people with disability to participate in a wide variety of sports and games. Existing sports can be improved to accommodate persons with disability, or a new game can be invented with persons with disability in mind, for instance, playing football using sticks, wheelchairs, and other games like racing, tennis, and basketball.
Home automation helps disabled people to live independently hence they opt to stay at home rather than move to a healthcare facility. For example, recorded audio messages and automated prompts. These includes switching off the oven, reminders to lock the door and kitchen implements with large grips for persons with arthritis or weakness in their hands. Medication dispensers with alarms to remind people to take medication. In entertainment, closed captioning enables people with hearing problems to enjoy television viewing and movies, adaptive switches enables children with impaired motor skills to effectively play with their toys. Some tools aid in daily activities for example eating and dressing. These include adapted utensils and costumed designed shower stalls and toilet seats. Alternative communication allows a child who is not able to speak or their speech not understood to communicate properly. These includes computers, picture boards, and communication software (Carver et al. 2016).
Positioning devices are used to support physically disabled person to remain in a normal position without becoming exhausted these includes modifiable tables, chairs straps, and wedges. Vehicles can also be modified to make it easier for disabled individuals to operate them for instance, foot pedals can be elevated or substituted with hand-controlled ones.
Prosthetics are used to replace a body part that is missing. These are biomechatronics that assist and enhance motor control following a disease, injury or defect. This technology enables the use of mechanical devices with skeletal, muscular and nervous systems to help in motor control. These include artificial eyes, limbs, hearing aids, and dentures. Also, individuals deprived of stimulation of senses can use sensory assistance or neurological assistive devices.
Assistive technology enables persons with disability to participate entirely in all aspects of life, increasing their opportunities and capabilities. It promotes independence and full control for persons with disability. Hence this leads to improvements in many fields including learning, motor, social, and cognitive and communication. It also lightens help care practitioners load.
The social model of disability contends for the removal of barriers or elements of the social structure which limits people with disability. Disabled people can fully participate in mainstream society just like other people hence enabling independence, equality in society and efficient control of their lives. As cited above, these barriers can be inaccessible buildings, unfavorable transport, lack of access to information, laid back stereotype, prejudice, rigid organizational practices, and procedures. With these adjustments, the load to caregivers reduces due to the creation of an enabling environment for people with disabilities (Thompson 2016).
Carver, J., Ganus, A., Ivey, J. M., Plummer, T., & Eubank, A. (2016). The impact of mobility assistive technology devices on participation for individuals with disabilities. Disability and Rehabilitation: Assistive Technology, 11(6), 468-477.
Shinohara, K., & Wobbrock, J. O. (2016). Self-conscious or self-confident? A diary study conceptualizing the social accessibility of assistive technology. ACM Transactions on Accessible Computing (TACCESS), 8(2), 5.
Thompson, L. S. C. A. (2016). Moving Beyond the Limits of Disability Inclusion: Using the Concept of Belonging Through Friendship to Improve the Outcome of the Social Model of Disability. World Academy of Science, Engineering and Technology, International Journal of Social, Behavioral, Educational, Economic, Business and Industrial Engineering, 10(5), 1454-1457.
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Although rare, the American bison has been saved from extinction. Once, vast herds of over a million bison grazed the vast prairies of western North America, often making migrations of several thousand kilometres to winter feeding grounds in the south. Bison were also widely killed for their skin and meat. Originally, bison also occurred extensively in mountain areas, and also in open forest and woodland.
These large grazing animals have well-developed senses of smell and hearing. The American bisons can run at up to 60 kmh (37 mph) and are also able to swim well, sometimes crossing rivers as wide as 1 km (0.6 mile).
Male bison are larger than the females of the species. Both sexes have sharp, curved horns, which stick out from the shaggy, brown hair on their heads.
While American bison may occasionally gather in herds of several hundred, they generally move around in small bands made up of a number of females and their offspring, including young bulls. Mature bulls either live alone or move in separate groups from the cows. During the mating season, in late summer, the males join the females. They fight for the females by ramming each other head-on.
Distribution: Patches of western Canada and central United States.
Habitat: Prairie and woodland.
Size: 2.1 - 3.5 m (7 - 11.5 ft); 350 - 1000 kg (770 - 2200 lb). Height at shoulder 1.5 - 2m (5 - 6.5 ft).
Maturity: 1 - 2 years.
Breeding: Single young born in spring every 1 or 2 years.
Life span: 40 years.
Status: Lower risk.
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Microbiology refers to the study of microorganisms which encompasses bacteriology, mycology, parasitology, and virology. Several fields of microbiology including medical microbiology, food microbiology, and clinical microbiology rely on the ability to culture and manipulate microbes. In each of these fields, antibiotics serve as an extremely useful tool to help culture or characterize bacteria. In research settings, antibiotics are frequently used as selection agents to select for resistant bacteria in transformation and transduction procedures. In medical settings, antibiotics can be used to determine how effective an antibiotic may be in vivo or to track antibiotic resistance patterns in emerging antibiotic resistant pathogens. Antibiotics can also be used in food microbiology screening labs to inhibit the growth of non-pathogenic food borne bacteria.
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On May 2, a rare earthquake shook Michigan, rolling in at magnitude 4.2, the second strongest quake recorded in the state. Fortunately, although items fell off of shelves and windows vibrated, the state suffered no major destruction or injuries from the quake.
In contrast to the West Coast, where over 350 faults have been reliably identified in connection with plate boundaries, earthquakes in the eastern US often have mysterious origins. Deep and ancient faults underlie the sleeping giant of the eastern continent, but few of these faults faults have been definitively mapped.
Michigan's 30-something recorded earthquakes have left no evidence at the surface, but the fault probably lies 60 miles deep; however, the focus of May's earthquake is placed at 3.7 miles below the surface.
While eastern US earthquakes are much less common, they have the potential to be devastating. The older, stiffer crust in the eastern US transmits the energy more than does the younger West Coast crust, meaning that earthquakes there are felt over an area more than ten times larger than would an earthquake of equivalent magnitude hitting the West Coast.
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Fossils are among the most valuable sources of information about the Earth's history. They tell us about the organisms that lived on Earth from the time of the oldest fossils, about 3.8 billion years ago, to the present. By studying fossils we can learn not only about the creatures and plants of the distant past, but how they grew, what they ate, how they interacted, and many aspects of their behavior.
Fossils reveal many fascinating facts about the past, but they do a lot more. Do you own anything made out of plastic? Plastic comes from oil, which also provides gasoline, gas heat, and many other necessities of modern life. Fossils are one of the most useful aids to finding oil, because oil tends to accumulate in the pores of particular rock layers. Rocks of different ages contain different fossils. Study of microscopic fossils brought up in chips of rock during drilling of wells has led to many major oil and gas discoveries. Also, the oil itself is derived from fossil remains of ancient organisms.
Study of fossils has led to important new understanding about how life evolved on earth and about diseases, both ancient and modern. Fossils also help us understand past climates, including ice ages and periods that were warmer than our present climate. Knowledge from the study of fossils is helping geoscientists understand global warming and its effects. By studying the catastrophic extinction of the dinosaurs and many other life forms at the end of the Cretaceous Period, geoscientists have gained insight into the evolutionary implications of impacts by extraterrestrial objects. Investigating the physical and chemical characteristics of fossil organisms that lived during times of drastic climatic change helps us understand the implications of the changes we are making in our own environment.
Information about Earth history, practical help in finding energy resources, and information that helps us anticipate the effects of possible environmental changes are not the only benefits derived from fossils. Fossils are beautiful. Many thousands of people collect, buy, sell, and trade fossils all over the world. Some jewelry and furniture are made from fossils, and many stone buildings are made from stone that is composed largely of fossils. Many people collect fossils simply because they are beautiful, but others do so because fossils tell fascinating stories. Neither Barney the Dinosaur nor Jurassic Park would exist if there were no fossils.
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We are destroying our natural spaces with pollution. Our shorelines are littered with debris that wash ashore from the oceans.
When the earthquake struck Japan in 2011, the destruction ended up being washed into the ocean. Eight tonnes of tsunami debris washed onto the beaches of Vancouver Island in a single year. Source: Vancouver Aquarium
Plastics break down into tinier fragments, and pose a threat to the local environment. Effects include:
• Birds ingesting the plastics, mistaking it as food - leading to starvation
• Animals impeded by debris to form their habitats along the shoreline
• Toxins leeching into the sand and eventually waterways
Our solution is a 3d printed robot platform for debris retrieval, exploration, and mobile sensor monitoring.
It's more than just the robot. Two different groups of people, makers and environmentalists, working together and applying their skills to the problem. Volunteers help out to test the robot during Field Tests. Engaging with the public during these tests is valuable to gain their experience on how robots can help improve the environment. Read more about this in our project log about a Field Test with the Great Canadian Shoreline Cleanup.
Debris retrieval is focused on tiny trash 10-50mm. An example is a plastic bottle cap. The end-effector scoop on the robot has a filter to let sand fall through, and the debris to remain. Using semi-autonomous behaviours, the robot will be more efficient at collecting this size of debris than by hand. Read more about our work on autonomous routines here, and magnetometer navigation.
Deploying this as a tool for Citizen Science (see log here) enables us to navigate to less accessible areas. Thus far we have tested up to 150m. We have used the GPS data from the robot to gain valuable insights about the location of debris, read more about it here.
During our baseline testing, the robot was able to collect and return 1 piece of debris in less than 90 seconds in 2.2 square meters while manually controlled.
For a 100 square meter portion of Cherry Beach that we frequented for Field Tests, we would estimate that using the robot for 1-2 hours would be able to collect 75% of the debris in that area. Our goal is to achieve this through semi-autonomous operation, rather than manual control.
Concept of Operations
Note: The command post is for relaying control commands received remotely from the internet, to the robot locally. Home base tent is in case a robot operator needs to be on site to monitor the initial deployment, and perform research experiments.
The arm and drive system are detachable using sliding dovetails, which lock in place with a printed key piece.
Base of the robot from underneath, without the arm and drive system connected. You can see the storage area for the battery on the left.
One of the challenges we encountered was reducing the amount of force applied to the motor - particularly when traversing rocky shorelines. See this project log for our force simulation. We designed a multi-material 3d printed wheel assembly with a flexible piece for suspension. This means we can use the existing motors - not needing to buy more expensive ones, and that this assembly can be replicated - no need to be shipping springs or other parts. Read more in this project log
To date on the systems architecture, we have the Nunchuck Controller working to control the Motor Controller onboard the robot, aka the Main Board. The Main Board has some of the sensors from the envisioned Sensors & Comms board to reduce complexity at this stage. We use the GPS Unit to communicate with the Main Board via serial to provide the robot's current latitude and longitude.
We are excited for the continued development of the system - in particular the...Read more »
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## Speed dating and graph theory
Suppose you are organising a speed dating event. You are given a graph $G = (S,P)$. The event consists of a number of rounds; in the $i$th round of the event, you must pick a subset $P_i\in P$ such that no $s\in S$ appears in two elements of $P_i$, and $P_i\cap P_j = \varnothing$ for all $j < i$. (That is, no person may be in two dates at the same time, and the same date may not occur in two rounds.) The event lasts for $R$ rounds, finishing when $\cup_{1\leq i\leq R} P_i = P$. (That is, when all possible dates have occurred). The ‘idle time’ is defined as $T = \sum_{1\leq i\leq R} \sum_{s\in S} 1(s \not\in p \forall p\in P_i)$.
For example, consider an event for $M$ straight men and $N$ straight women. Then $G$ is the complete bipartite graph $K_{M,N}$, and a simple solution is to have all the men seated around the inside of a ring-shaped table, all the women around the outside, and to have one set rotate around after each round while the other set stays still. In this case, $R = \max(M,N)$ and $T = R |M-N|$.
How can you choose the $P_i$ to minimise $T$?
## Doubling in backgammon
You are playing backgammon (wlog as white). (The rules of backgammon are explained more clearly here.) At any given point during a round, there are six possible outcomes of the round, with corresponding values:
• white wins by a backgammon ($+3n$)
• white wins by a gammon ($+2n$)
• white wins ($+n$)
• red wins ($-n$)
• red wins by a gammon ($-2n$)
• red wins by a backgammon ($-3n$)
At the start of a round, $n=1$ and both players have control of the doubling cube.
When it is Alice’s turn, she may offer a double if and only if she controls the doubling cube. If Alice offers a double, then Bob must accept or decline. If Bob declines, then Bob forfeits the round and Alice gains $n$ points. If Bob accepts, then Alice ceases to control the cube, Bob gains control of the cube, and $n$ is multiplied by 2.
(There are other rules.)
## The problem
Suppose that both you and your opponent can perfectly calculate the probability of each of the above outcomes.
• It is your turn. Should you offer a double?
• It is your opponent’s turn, and they have offered a double. Should you accept?
For a harder problem, suppose that you assign probabilities $p_i$ to each of the above outcomes, and your opponent assigns different probabilities $q_i$, but you know both the $p_i$ and the $q_i$.
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Rhythm, though one of the two most basic elements of music (the other being pitch), is also one of the most often misunderstood. Rhythm is frequently presented to students in the form of a hierarchy of note types, with a whole note at the top, two half notes beneath, four quarter notes below them, and so forth. The idea is to teach note values as an introduction to rhythm. But a note does not make a rhythm; it only makes a duration. It takes a sequence of durations to make a rhythm. This is because rhythm must have a beat, and one note cannot signify a beat. One duration could be any number of beats. For example, if I only hear one duration I don’t know if it is two beats at mm = 120, or one beat at mm = 60, or three beats at mm = 180. Without an established beat, there can be no rhythm. If a series of equal durations were to be played, one after the other with no pause in between, the problem would still persist. Indeed, different listeners could conceivably audiate any one of the beats I proposed, and all would be just as correct as the other. They could even audiate different meters, and again all be equally right. From this, we can see that one duration, or a series of identical durations is insufficient to establish beat or meter; therefore, a single duration or a series of identical durations does not qualify as a rhythm.
To qualify as a rhythm, there must be a series of durations that form a repeatable pattern with a clear beginning and a clear end that establishes both beat and meter. The problem with our series of identical durations is that it could begin or end anywhere—there is no way of knowing where it ends until the music comes to a final stop. Even worse, there is no way of establishing a pattern of strong and weak beats that would indicate a meter. Rhythms establish through patterns of differing durations exactly where each beat begins and where it ends, and where each metric beat is located—that is where each strong beat is amid the other weaker beats. Where all of this is clear, there is rhythm, and where there is rhythm, there is a beat hierarchy.
Recall the chart I described earlier—the one with a whole note at the top, divided into two half notes beneath, divided into four quarter notes beneath, and so on. Now we are ready to use this chart in a more effective way than teaching note durations. More effective, because the important issue is not that a whole note, tow half notes or four quarter notes all “add up” to four beats. None of these note types always gets any particular number of beats. The important issue is the relationship between the note types—four quarter notes occupy the same time span as two half notes or one whole note, regardless of how many beats each one gets. According to this, we understand that a piece in common time can be audited at the quarter note level, with quarter notes getting one beat, at the half note level, with half notes getting one beat at half the tempo, at the whole note level with whole notes getting one beat at one-quarter the tempo, or at the eighth note level with eighth notes getting one beat at twice the tempo.
As with melody, we can ask our students if the sound of passing cars on the road, or the sounds of birdsong is a rhythm. We know that one passing car cannot be a rhythm, because it is an isolated duration with no established beat or meter. Several cars passing by separated by long intervals of time pose the same problem, we cannot establish a beat from the time in between each event. Only when cars pass by close enough to each other to establish a beat pattern can we consider those sounds a rhythm. With bird song, it is typically repetitive and in patterns of pitches and durations which afford the listener to establish a beat and a meter; therefore, birdsong is rhythm. After teaching your students the necessity of sounds establishing beat an meter, see if they can fighre out what sounds are or are not rhythm. It is a good problem for them to struggle with. Make sure they can defend their conclusion.
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# Second moment of area
1. Dec 23, 2012
### Payam30
how do calculate send moment of area.
here is the exampel. I do understand the way one calculate Ixx and Iyy and Ixy. the smaller part has thikness t and the biggest part 3t.
I get :
if you place global coordinate system on the top you will get the position of center of gravity to CG=(3a/8,-a/4).
Now I do: define variable s which is the road
Ixx=∫y²dA=∫t(s sin30)²ds=t[(S³/3)(1/4)]=ta³/12
and we do the same with the right part. and we will get 3ta³/12
Adding these two gives ta³/3. but in the solution they have
(ta³/3)*(1/2)².
I dont get where (1/2)² is comming from.
This is not homework! This is an example from a exam!
Last edited: Dec 23, 2012
2. Dec 23, 2012
### Simon Bridge
well, sin(30)=1/2 ... which may give us a clue.
the first part gives: $$\frac{ta^3}{12}=\frac{ta^3}{3}\frac{1}{2^2}$$ - the second part gives: $$\frac{3ta^3}{12}=\frac{3ta^3}{3}\frac{1}{2^2}$$... adding them together gives: $$\frac{ta^3}{3}\frac{1}{2^2}+\frac{3ta^3}{3}\frac{1}{2^2}=\frac{4ta^3}{3}\frac{1}{2^2}$$... so the question is not so much where the (1/2)2 comes from - but where the factor of 4 went.
I see you have two beams length a, one of thickness t and the other of thickness 3t, which meet at an angle which is not specified on the diagram. There is a "30" on the diagram which I take to mean that one of the beams makes and angle of 30 degrees to something but the "something" is not specified.
From your analysis I'm guessing you are trying to find the polar moment Jxx by integrating over each beam and adding them - thus: you have oriented your coordinate axis so the y-axis is parallel to the bending force?
if $dA=t.ds$ and $y=s\sin(30)$ then I guess the beams meet at 90 degrees to each other and the light (thickness 1t) beam forms an angle of 30 degrees to the horizontal. It this correct?
I'd rather not guess!
I think we'd need this information.
3. Dec 24, 2012
### Payam30
Hi simon! thank you for answer.
Yes the right answer according to solution is 4ta³/12*(1/2²). sorry if I wrote it wrong at the first place. my bad.
I still don't get where (1/2) ² is comming from. I write it again how I do it: yes the angles is 30 on the both sides and is not 90 at the top No! y-axis is parallell to
$∫y².dA=∫t.(s \sin30)²ds=[t.(s³/3 )sin30²]=[ta³/12]$. so (1/2) just went.
$∫y².dA=∫3t.(s \sin30)²ds=[3(t.s³/3 )sin30²]=[3ta³/12]$. so (1/2) just went.
ta³/3 which is not correct . the answer is :
$(ta³/3).(1/2²)$
4. Dec 24, 2012
### Simon Bridge
sin2(30) = (1/2)2 like I said - look carefully at the derivation I showed you at the start of post #2 ... you have exactly the same result they have only you have multiplied out the known values... that's where the (1/2) "just went". i.e. $$\int y^2dA = \int (s\sin(30))^2tds = \int \frac{ts^2}{2^2}ds$$... keep the $(1/2)^2$ like that instead of evaluating it and see what happens.
Last edited: Dec 24, 2012
5. Dec 26, 2012
### Payam30
No. Actully my last result is :
$$\frac{ts^3}{3}$$
while their is :
$$\frac{ts^3}{3}\frac{1}{4}$$
6. Dec 26, 2012
### Simon Bridge
Which is it?
What you need to do is identify and report the correct solution and also go back over your integration showing the intermediate steps.
You should also provide the information requested in post #2.
7. Dec 26, 2012
### Payam30
they are all the same.
what i get from my integration is :
ta^3 /3
and the slution in the exam says that the result should be:
(4ta^3 / 12)(1/4) which is (ta^3/3)(1/4) which is (ta^3/3)(1/2^2) so all of the same the problemnis that just get ta^3/3which is 4 times bigger than that in the solution.
8. Dec 27, 2012
### Simon Bridge
The three quoted values in post #6 are not all the same - please reread carefully: the last one has an s in it which is not present in the other two ... I included the first two because the form of the solution appears to be contributing to the confusion.
In post #7 you are telling me something different for your answer as well.
How did you get $\frac{ta^3}{3}$ ?
In post #6 I have asked to see this working - do you not want to supply it?
In post #2 I asked about the angle between the two beams - the working shown in post #1 suggests that both beams may be 30deg from the horizontal - is that correct? As drawn in post #1 there is actually not enough information to solve the problem.
Please be more careful with your typing - typos in the math just add to the confusion.
9. Dec 28, 2012
### Payam30
10. Dec 28, 2012
### Simon Bridge
OK - now I see.
The model answer is just using a rule - knowing the second moment for a beam and how to combine them. That way they could do it in one step. Simple addition, that you used, does not always hold.
I think you should look at the examples in the wikipedia article on "second moment of inertia" to see what I mean then go through the calculation again without leaving out any steps. Meantime, I'll see if I can't produce a useful model answer for you.
11. Dec 29, 2012
### Payam30
I will be very thankfull if you could explain it to me. Whatever I do I cant get where the (1/2)² is coming from. Will you do me some calculations? please? I have an exam soon and it is a real pain in the ***.
12. Jan 1, 2013
### Simon Bridge
Sorry for the delay - I've been somewhat busy myself.
Preparing for exams over xmas/new-year is a bummer!
Anyhow - I had a go looking for the 1-step approach used by the model answer and got nowhere. Someone with more recent experience should do better ... I can only conclude that the examiner was expecting people to use a remembered result from class. You should look through your notes although it is possible that the particular method was not used this year.
It is also possible that the model answer or the question is somehow in error :)
Anyway - the way to check would be to use the transformations.
Divide the area into two rectangular areas. Let area 1 be the thin rectangle and area 2 be the fat one.
Use the following observations:
(iirc)
For a rectangular area with extent a along the x axis and t along the y axis, the CG moments are: $$J_{xx}=\frac{at^3}{12}\; ; \; J_{yy}=\frac{a^3t}{12}\; ; \; J_{xy}=0$$
After a rotation angle $\phi$ in the x-y plane: $$[J_{xx}]_{rot}=\frac{1}{2}(J_{xx}+J_{yy})+\frac{1}{2}(J_{xx}-J_{yy})\cos(2\phi)-J_{xy}\sin(2\phi)$$ ...etc.
... but all that is for an axis through the centroid. If the axis is a distance d from the centroid, then you use the parallel axis theorum: $$J_{xx}=[J_{xx}]_{CG}+Ad^2$$ ... for you, for $J_{xx}$, $A=4ta$ and $d=a/4$ (if the x-axis passes through the apex of the triangle formed by the beams.)
... you'll have to modify for $J_{yy}$ but you already know the center of mass coordinates.
There is going to be some approximation here - you'll have to use your judgement. I was working with the origin through the center of one end of each - which means there is an overlap of area $3t^2/4$ which I hope is small. You may want to use a different geometry.
For the small area I get: $$J_{xx}=\frac{3}{4}\frac{at^3}{12}+\frac{ta^3}{12}$$ ... which seems suggestive. It's been a while since I had to do these though so check yourself.
13. Jan 2, 2013
### Payam30
Amazing. thank you very much Simon!!
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Each year, experts from the US Food and Drug Administration, World Health Organization, and the Centers for Disease Control and Prevention study virus samples collected from 136 national influenza centers in 106 countries to determine what three strains will be the most common during that particular season. These strains are then included in the annual vaccine.
According to the Centers for Disease Control and Prevention, older adults, pregnant women, young children, and people with certain health conditions like asthma, diabetes, and heart disease, are at greater risk for complications if they get the flu. Many individuals are eligible to receive the flu shot at little or no cost. Go to flu.gov for more information.
Nevertheless, it is possible to get the flu after being vaccinated. The vaccination works best on those who are young and healthy, but older adults and those with compromised immune systems may not be as well-protected. Additionally, the body takes approximately 2 weeks to build up immunity from the vaccine. People exposed before or within the two-week time period could possibly get the flu. And lastly, because the vaccination only covers 3 strains, it does not protect against all flu viruses.
The viruses in the flu vaccinated have been inactivated, so you will not get the flu from the shot. According to the CDC, "The risk of serious harm or death is extremely small. However, a vaccine, like any medicine, may rarely cause serious problems, such as severe allergic reactions. Almost all people who get influenza vaccine have no serious problems from it." The benefits far outweigh the risks.
Certain people, however, should not be vaccinated without consulting a physician first. These include:
- Those who have severe allergies to chicken eggs
- People suffering from an illness with a fever
- People with a history of Guillain-Barre Syndrome
- Children younger than 6 months of age
Research suggests that intakes of 200 mg/day or more of Vitamin C will not reduce the incidence of illness. However, it may be helpful for those with marginal vitamin C statuses, such as elderly and smokers, or those exposed to extreme physical exercise or cold environments. Vitamin C may shorten the duration of the common cold, but it appears that taking it after the onset of the cold or flu is not beneficial.
To Exercise or Not to Exercise?
According to the National Institutes of Health, exercise has not been shown in research to prevent colds or the flu. However, exercise, along with a proper diet, can improve the immune system by helping the disease-fighting white blood cells in the body move from the organs into the bloodstream. This helps decrease your chances of getting a cold or the flu.
If you happen to get any sort of seasonal illness, whether it's a cold or flu, you should not exercise if you have chest congestion, hacking cough, upset stomach, body aches, fatigue, or widespread muscle aches. However, if you have only mild symptoms, such as a runny nose, nasal congestion, or sneezing, exercise could help by opening nasal passages. During this time, however, you should reduce the intensity and length of your exercise.
How to Prevent Sickness During Flu Season
There are several things you can do to prevent illness.
- Ensure that your immune system is in tip top condition - eat a healthy diet, get regular physical activity, manage stress and get plenty of rest (at least 7 hours of sleep a night for adults).
- At work or school, wash your hands often, especially after working with someone who may be ill and before eating.
- If you can't get to a sink on a regular basis, use an alcohol based hand sanitizer - but be aware that hand sanitizer is not as effective as washing your hands.
Mandy Seay is a bilingual registered and licensed dietitian who holds both a bachelor's degree in nutrition and in journalism. After gaining 30 pounds while living abroad, Mandy worked to lose the weight and regain her health. It was here that she discovered her passion for nutrition and went on to pursue a career as a dietitian. Mandy currently works as a nutrition consultant and freelance writer in Austin, Texas, where she specializes in diabetes, weight management and general and preventive nutrition. She recently published her first book, Your Best Health, a personalized program to losing weight and gaining a healthy lifestyle. Please visit Mandy's website at Nutritionistics.com.
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# Thread: What to do next? (derivative)
1. ## What to do next? (derivative)
Hi, I'm beginner to calculus and I forgot my algebra because I stop my study for years. Now I don't know what next after I plug the value to the formula of quotient rule in derivate.
This is the given.
$\displaystyle y = \frac{x^3}{(x^2+4)^2}$
Then I think this is the derivative of the given function, but how to solve or factor or simplify this?
$\displaystyle = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}$
If you can show me the full solution then explain what process did you do, it will help me alot. Thanks
2. ## Re: What to do next? (derivative)
Originally Posted by avahdon
Hi, I'm beginner to calculus and I forgot my algebra because I stop my study for years. Now I don't know what next after I plug the value to the formula of quotient rule in derivate.
This is the given.
$\displaystyle y = \frac{x^3}{(x^2+4)^2}$
Then I think this is the derivative of the given function, but how to solve or factor or simplify this?
$\displaystyle = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}$
If you can show me the full solution then explain what process did you do, it will help me alot. Thanks
if
$f(x) = \dfrac{g(x)}{h(x)}$
and
$f^\prime(x)=\dfrac{df}{dx}(x)$
then
$f^\prime(x)=\dfrac{g^\prime(x)h(x)-g(x)h^\prime(x)}{(h(x))^2}$
so in this case
$g(x)=x^3$
$g^\prime(x)=3x^2$
$h(x)=(x^2+4)^2$
$h^\prime(x)=2(x^2+4)2x$
you should be able to put all the pieces together from here.
3. ## Re: What to do next? (derivative)
Originally Posted by romsek
if
$f(x) = \dfrac{g(x)}{h(x)}$
and
$f^\prime(x)=\dfrac{df}{dx}(x)$
then
$f^\prime(x)=\dfrac{g^\prime(x)h(x)-g(x)h^\prime(x)}{(h(x))^2}$
so in this case
$g(x)=x^3$
$g^\prime(x)=3x^2$
$h(x)=(x^2+4)^2$
$h^\prime(x)=2(x^2+4)2x$
you should be able to put all the pieces together from here.
I already solve the derivative and I have this $\displaystyle = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}$
my question now is how to simplify, solve, or factor this? because my teacher don't accept this answer
4. ## Re: What to do next? (derivative)
Originally Posted by avahdon
Hi, I'm beginner to calculus and I forgot my algebra because I stop my study for years. Now I don't know what next after I plug the value to the formula of quotient rule in derivate.
This is the given.
$\displaystyle y = \frac{x^3}{(x^2+4)^2}$
Then I think this is the derivative of the given function, but how to solve or factor or simplify this?
$\displaystyle = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}$
If you can show me the full solution then explain what process did you do, it will help me alot. Thanks
One thing you should be able to see that there is a term of the form $\displaystyle x^2+ 4$ in both terms in the numerator and one in the denominator so you can cancel those:
$\displaystyle \frac{(x^2+ 4)(3x^2)- (x^3)(2x)}{(x^2+ 4)^3}$
(You did recognize that $\displaystyle ((x^2+ 4)^2)^2$ is $\displaystyle (x^2+ 4)^4$, right?)
Now multiply out $\displaystyle (x^2+ 4)(3x^3)$ and $\displaystyle (x^3)(2x)$ and subtract.
(I strongly recommend that you start reviewing algebra and trig.)
5. ## Re: What to do next? (derivative)
Hello, avahdon!
$\displaystyle y \:=\: \frac{x^3}{(x^2+4)^2}$
. . $\displaystyle y' \:=\: \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}$
How to simplify this?
You have: .$\displaystyle y' \;=\;\frac{3x^2(x^2+4)^2 - 4x^4(x^2+4)}{(x^2+4)^4}$
Factor: .$\displaystyle y' \;=\;\frac{x^2(x^2+4)\big[3(x^2+4) - 4x^2\big]}{(x^2+4)^4}$
Reduce: .$\displaystyle y' \;=\;\frac{x^2\big[3x^2+12 - 4x^2\big]}{(x^2+4)^3}$
Simplify: .$\displaystyle y' \;=\;\frac{x^2(12-x^2)}{(x^2+4)^3}$
6. ## Re: What to do next? (derivative)
Originally Posted by HallsofIvy
One thing you should be able to see that there is a term of the form $\displaystyle x^2+ 4$ in both terms in the numerator and one in the denominator so you can cancel those:
$\displaystyle \frac{(x^2+ 4)(3x^2)- (x^3)(2x)}{(x^2+ 4)^3}$
(You did recognize that $\displaystyle ((x^2+ 4)^2)^2$ is $\displaystyle (x^2+ 4)^4$, right?)
Now multiply out $\displaystyle (x^2+ 4)(3x^3)$ and $\displaystyle (x^3)(2x)$ and subtract.
(I strongly recommend that you start reviewing algebra and trig.)
thanks, can you tell me which topic should i review in algebra and trig?
Originally Posted by Soroban
Hello, avahdon!
You have: .$\displaystyle y' \;=\;\frac{3x^2(x^2+4)^2 - 4x^4(x^2+4)}{(x^2+4)^4}$
Factor: .$\displaystyle y' \;=\;\frac{x^2(x^2+4)\big[3(x^2+4) - 4x^2\big]}{(x^2+4)^4}$
Reduce: .$\displaystyle y' \;=\;\frac{x^2\big[3x^2+12 - 4x^2\big]}{(x^2+4)^3}$
Simplify: .$\displaystyle y' \;=\;\frac{x^2(12-x^2)}{(x^2+4)^3}$
thanks.
7. ## Re: What to do next? (derivative)
Alternatively, use logarithmic differentiation.
y=x^3/(x^2+4)^2, so
ln(y) = ln(x^3) - ln(x^2+4)^2 = 3ln(x)-2ln(x^2+4)
Differentiating both sides, you have
y'/y = 3/x - 2*1/(x^2+4)*2x = 3/x -4x/(x^2+4) = [3(x^2+4)-4x^2]/[x(x^2+4)] = [-x+12]/[x(x^2+4)]
Now, solving for y', you just multiply both sides by y:
y' = [-x+12]/[x(x^2+4)] * x^3/(x^2+4)^2 = (12-x)/(x^2+4) * x^2/(x^2+4)^2 = (12x^2-x^3)/(x^2+4)^3. //
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0
# Is 33 plus 0 33 a distributive property?
Updated: 9/25/2023
Wiki User
8y ago
Unfortunately, the browser used by Answers.com for posting questions is incapable of accepting mathematical symbols. This means that we cannot see the mathematically critical parts of the question. We are, therefore unable to determine what exactly the question is about and so cannot give a proper answer to your question. However, it would appear that this is NOT an example of the distributive property but, instead, the identity property of 0 with respect to addition.Please edit your question to include words for symbols and resubmit.
Wiki User
8y ago
Wiki User
8y ago
Even though the question is not clear, it is very definitely NOT an example of the distributive property.
Earn +20 pts
Q: Is 33 plus 0 33 a distributive property?
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Related questions
### Which property is illustrated in this problem 33 plus 0 equals 33?
The fact that 0 is the additive identity over for integers or rationals or reals.
### What is distributive property in math terms?
well it means a number times another is equal to a number plus another= 4x5=20+0 * * * * * No. The correct answer is as follows: The distributive property of multiplication over addition states that a*(b + c) = a*b + a*c that is, the multiplication of the bracket by a can be distributed over the elements inside the bracket.
### What is 7(n plus 4)21?
0 = 7(n + 4)21Use distributive property to multiply 7 by n and 4.0 = (7n + 28)21Use Distributive property to multiply 21 by 7n and 28.0 = 147n + 588Subtract 147n from both sides of the equation.-147n = 588Divide the entire equation by -147n.n = -4
### What property is 18 plus 0 equals 0 plus 18?
This is the Commutative property or law.
### What is the property of x plus 0?
The Identity Property
### What is the property of -10 plus 0 equals -10?
It is the property that 0 is the identity for addition.
### Which property is represented 15 plus 0?
It is the Identity Property
### What is the sum of 2X plus 1 and its opposite?
Zero. The sum of anything and it's opposite is zero, that's how an opposite is defined. In this case, the opposite of 2x + 1 is -(2x + 1) = -2x - 1 by the distributive property. Adding like terms, 2x + -2x + 1 +-1 = 0 + 0 = 0.
### What property is -4 plus 0 -4 in Algebra?
It is the additive identity property of the number 0.
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Question 3d744
Jun 19, 2017
The final temperature of the copper is 18.7 °C.
Explanation:
This is a typical calorimetry question.
There are two heat transfers involved:
$\text{heat lost by Cu + heat gained by water} = 0$
$\textcolor{w h i t e}{m m m m} {q}_{1} \textcolor{w h i t e}{m m l l} + \textcolor{w h i t e}{m m m m} {q}_{2} \textcolor{w h i t e}{m m m m m} = 0$
color(white)(mm)m_1C_1ΔT_1 color(white)(m)+color(white)(mm) m_2C_2ΔT_2 color(white)(mmm)= 0
Let's calculate each heat separately.
q_1 = m_1C_1ΔT_1 = 250. color(red)(cancel(color(black)("g"))) × "0.385 J"·color(red)(cancel(color(black)("g"^"-1")))"°C"^"-1" × (T_text(f) - "100 °C") = "96.25 J·°C"^"-1"(T_text(f) - "100 °C") = 96.25T_text(f) color(white)(l)"J·°C"^"-1" - "9625 J"
q_2 = m_2C_2ΔT_2 = 500 color(red)(cancel(color(black)("g"))) × "4.184 J"·color(red)(cancel(color(black)("g"^"-1")))"°C"^"-1" × (T_text(f) - "15 °C") = "2092 J·°C"^"-1"(T_text(f) - "15 °C") = 2092T_text(f) color(white)(l)"J·°C"^"-1" - "31 380 J"#
Now, we add the two heats and combine like terms.
${q}_{1} + {q}_{2} = 96.25 {T}_{\textrm{f}} \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{J")))·"°C"^"-1" - "9625" color(red)(cancel(color(black)("J"))) + 2092T_text(f) color(red)(cancel(color(black)("J")))·"°C"^"-1" - "31 380" color(red)(cancel(color(black)("J}}}} = 0$
$2188 {T}_{\textrm{f}} \textcolor{w h i t e}{l} \text{°C"^"-1" - "41 005} = 0$
${T}_{\textrm{f}} = \text{41 005"/("2188 °C"^"-1") = "18.7 °C}$
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Lesson Objectives
• Learn how to solve basic trigonometric equations
• Learn how to solve trigonometric equations with half-angles
## How to Solve Trigonometric Equations with Half-Angles
Over the course of the last few lessons, we learned how to solve trigonometric equations using linear methods, factoring, and squaring. Here, we will take the next step and learn how to work with trigonometric equations that have half-angles.
### Unit Circle
The unit circle will be given here for reference.
### Solving Trigonometric Equations with Half-Angles
In some cases, we will be asked to solve a trigonometric equation with a half-angle. Sometimes, we will need to use our half-angle identities. In other cases, this won't be necessary. Let's look at a few examples.
Example #1: Solve each equation over the interval $[0, 2π)$. $$\text{sin}\hspace{.1em}\frac{β}{2}=\sqrt{2}- \text{sin}\hspace{.1em}\frac{β}{2}$$ Let's first consider our interval written as an inequality: $$0 ≤ β < 2π$$ Since we have β/2, let's divide each part by 2: $$0 ≤ \hspace{.15em}\frac{β}{2}< π$$ Now, let's revisit our equation and find all values of β/2 over the interval $[0, π)$: $$\text{sin}\hspace{.1em}\frac{β}{2}=\sqrt{2}- \text{sin}\hspace{.1em}\frac{β}{2}$$ Add sin β/2 to both sides: $$2\text{sin}\hspace{.1em}\frac{β}{2}=\sqrt{2}$$ Divide both sides by 2: (Note: The angle is β/2) $$\text{sin}\hspace{.1em}\frac{β}{2}=\frac{\sqrt{2}}{2}$$ We want to find all values of β/2 over the interval $[0, π)$ that satisfy our equation. $$\text{sin}\hspace{.1em}\frac{π}{4}=\frac{\sqrt{2}}{2}$$ Sine is also positive in quadrant II.
What angle has a reference angle of $\frac{π}{4}$ or 45° in quadrant II? $$π - \frac{π}{4}=\frac{3π}{4}$$ $$\text{sin}\hspace{.1em}\frac{3π}{4}=\frac{\sqrt{2}}{2}$$ Our last step is to set β/2 equal to each and solve for β. $$\frac{β}{2}=\frac{π}{4}$$ $$\text{or}$$ $$\frac{β}{2}=\frac{3π}{4}$$ Let's start with the top equation. $$\frac{β}{2}=\frac{π}{4}$$ Multiply both sides by 2: $$β=\frac{π}{2}$$ Let's now work on the bottom equation. $$\frac{β}{2}=\frac{3π}{4}$$ Multiply both sides by 2: $$β=\frac{3π}{2}$$ Our solutions for β in our given interval: $$\left\{\frac{π}{2}, \frac{3π}{2}\right\}$$ Let's now look at an example that uses a half-angle identity.
Example #2: Solve each equation over the interval $[0, 2π)$. $$-\text{cos}\hspace{.1em}θ=-2 + 3\text{sin}\hspace{.1em}\frac{θ}{2}$$ Let's replace sin θ/2 using the half-angle identity: $$\text{sin}\hspace{.1em}\frac{A}{2}=\pm \sqrt{\frac{1 - \text{cos A}}{2}}$$ $$-\text{cos}\hspace{.1em}θ=-2 + 3\text{sin}\hspace{.1em}\frac{θ}{2}$$ $$-\text{cos}\hspace{.1em}θ=-2 \pm 3\sqrt{\frac{1 - \text{cos θ}}{2}}$$ Let's add 2 to both sides of the equation: $$2 -\text{cos}\hspace{.1em}θ=\pm 3\sqrt{\frac{1 - \text{cos θ}}{2}}$$ Square both sides: $$(2 -\text{cos}\hspace{.1em}θ)^2=\left(\pm 3\sqrt{\frac{1 - \text{cos}\hspace{.1em}θ}{2}}\right)^2$$ $$4 - 4\text{cos}\hspace{.1em}θ + \text{cos}^2 θ=9 \cdot \frac{1 - \text{cos}\hspace{.1em}θ}{2}$$ Multiply both sides by 2: $$8 - 8\text{cos}\hspace{.1em}θ + 2\text{cos}^2 θ=9(1 - \text{cos}\hspace{.1em}θ)$$ Distribute the 9 on the right-hand side: $$8 - 8\text{cos}\hspace{.1em}θ + 2\text{cos}^2 θ=9 - 9\text{cos}\hspace{.1em}θ$$ Subtract 9 away from each side and add 9 cos θ to both sides: $$-1 + \text{cos}\hspace{.1em}θ + 2\text{cos}^2 θ=0$$ Rearrange into $ax^2 + bx + c=0$: $$2\text{cos}^2 θ + \text{cos}\hspace{.1em}θ - 1=0$$ Factor the left-hand side: $$(2\text{cos}\hspace{.1em}θ - 1)(\text{cos}\hspace{.1em}θ + 1)=0$$ Use the zero-factor property: $$2\text{cos}\hspace{.1em}θ - 1=0$$ $$\text{or}$$ $$\text{cos}\hspace{.1em}θ + 1=0$$ Let's solve the top equation first: $$2\text{cos}\hspace{.1em}θ - 1=0$$ Add 1 to each side, then divide both sides by 2: $$\text{cos}\hspace{.1em}θ=\frac{1}{2}$$ $$θ=\frac{π}{3}, \frac{5π}{3}$$ Let's now solve the bottom equation: $$\text{cos}\hspace{.1em}θ + 1=0$$ $$\text{cos}\hspace{.1em}θ=-1$$ $$θ=π$$ Our solutions for θ in the given interval: $$\left\{\frac{π}{3}, π, \frac{5π}{3}\right\}$$ Note: when we square both sides of an equation, we need to check our solutions in the original equation. To keep this tutorial shorter, we will only show the check when we have an extraneous solution. Let's look at one more example using a half-angle identity.
Example #3: Solve each equation over the interval $[0, 2π)$. $$\sqrt{3}\text{cos}\hspace{.1em}\frac{θ}{2}=1 + \text{cos}\hspace{.1em}θ$$ Let's replace cos θ/2 using the half-angle identity: $$\text{cos}\hspace{.1em}\frac{θ}{2}=\pm \sqrt{\frac{1 + \text{cos}\hspace{.1em}θ}{2}}$$ $$\pm \sqrt{3}\sqrt{\frac{1 + \text{cos}\hspace{.1em}θ}{2}}=1 + \text{cos}\hspace{.1em}θ$$ Square both sides: $$\frac{3(1 + \text{cos}\hspace{.1em}θ)}{2}=1 + 2\text{cos}\hspace{.1em}θ + \text{cos}^2 θ$$ Multiply both sides by 2: $$3(1 + \text{cos}\hspace{.1em}θ)=2 + 4\text{cos}\hspace{.1em}θ + 2 \text{cos}^2 θ$$ Distribute the 3 on the left side: $$3 + 3\text{cos}\hspace{.1em}θ=2 + 4\text{cos}\hspace{.1em}θ + 2 \text{cos}^2 θ$$ Let's move all terms to the right and place in the form: $$0=ax^2 + bx + c$$ $$0=2\text{cos}^2 θ + \text{cos}\hspace{.1em}θ - 1$$ Flip Sides: $$2\text{cos}^2 θ + \text{cos}\hspace{.1em}θ - 1=0$$ Factor the left-hand side: $$(2\text{cos}\hspace{.1em}θ - 1)(\text{cos}\hspace{.1em}θ + 1)=0$$ Use the zero-product property: $$2\text{cos}\hspace{.1em}θ - 1=0$$ $$\text{or}$$ $$\text{cos}\hspace{.1em}θ + 1=0$$ Let's solve the top equation first: $$2\text{cos}\hspace{.1em}θ - 1=0$$ Add 1 to each side, then divide both sides by 2: $$\text{cos}\hspace{.1em}θ=\frac{1}{2}$$ $$θ=\frac{π}{3}, \frac{5π}{3}$$ Let's now solve the bottom equation: $$\text{cos}\hspace{.1em}θ + 1=0$$ Subtract 1 from each side of the equation: $$\text{cos}\hspace{.1em}θ=-1$$ $$θ=π$$ As we mentioned above and in previous tutorials, when we square both sides of an equation, it is possible to obtain extraneous solutions. Let's check our proposed solutions in the original equation to see if they work: $$\sqrt{3}\text{cos}\hspace{.1em}\frac{θ}{2}=1 + \text{cos}\hspace{.1em}θ$$ Replace θ with each proposed solution: Let's start with $\frac{π}{3}$: $$\sqrt{3}\text{cos}\hspace{.1em}\frac{\large{\frac{π}{3}}}{2}=1 + \text{cos}\hspace{.1em}\frac{π}{3}$$ $$\sqrt{3}\text{cos}\hspace{.1em}\frac{π}{6}=1 + \text{cos}\hspace{.1em}\frac{π}{3}$$ $$\sqrt{3}\cdot \frac{\sqrt{3}}{2}=1 + \frac{1}{2}$$ $$\frac{3}{2}=\frac{2}{2}+ \frac{1}{2}$$ $$\frac{3}{2}=\frac{3}{2}$$ $\frac{π}{3}$ is a valid solution.
Let's now check $\frac{5π}{3}$: $$\sqrt{3}\text{cos}\hspace{.1em}\frac{\large{\frac{5π}{3}}}{2}=1 + \text{cos}\hspace{.1em}\frac{5π}{3}$$ $$\sqrt{3}\text{cos}\hspace{.1em}\frac{5π}{6}=1 + \text{cos}\hspace{.1em}\frac{5π}{3}$$ $$\sqrt{3}\cdot -\frac{\sqrt{3}}{2}=1 + \frac{1}{2}$$ $$-\frac{3}{2}=\frac{2}{2}+ \frac{1}{2}$$ $$-\frac{3}{2}=\frac{3}{2}$$ $\frac{5π}{3}$ is not a valid solution. Lastly, let's check $π$: $$\sqrt{3}\text{cos}\hspace{.1em}\frac{π}{2}=1 + \text{cos}\hspace{.1em}π$$ $$\sqrt{3}\cdot 0=1 + (-1)$$ $$0=0$$ $π$ is a valid solution. $$\left\{\frac{π}{3}, π\right\}$$
#### Skills Check:
Example #1
Solve each equation for 0 ≤ θ < 2π $$-3=-4\text{cos}\hspace{.1em}\frac{θ}{2}+ \text{cos}\hspace{.1em}θ$$
A
$$\left\{\frac{5π}{6}\right\}$$
B
$$\left\{0\right\}$$
C
$$\left\{π\right\}$$
D
$$\left\{0, \frac{π}{2}\right\}$$
E
$$\left\{\frac{7π}{6}\right\}$$
Example #2
Solve each equation for 0 ≤ θ < 2π $$2\text{cos}\hspace{.1em}θ + 4\text{sin}\hspace{.1em}\frac{θ}{2}=3$$
A
$$\left\{\frac{π}{3}\right\}$$
B
$$\left\{\frac{2π}{3}, \frac{4π}{3}\right\}$$
C
$$\left\{\frac{π}{3}, \frac{5π}{3}\right\}$$
D
$$\left\{\frac{11π}{6}\right\}$$
E
$$\text{No Solution}$$
Example #3
Solve each equation for 0 ≤ θ < 2π $$\text{cos}\hspace{.1em}θ=3\text{sin}\hspace{.1em}\frac{θ}{2}+ 2$$
A
$$\left\{\frac{2π}{3}, \frac{4π}{3}\right\}$$
B
$$\left\{\frac{π}{3}, π, \frac{5π}{3}\right\}$$
C
$$\left\{π, \frac{3π}{2}\right\}$$
D
$$\text{No Solution}$$
E
$$\left\{\frac{π}{6}, \frac{11π}{6}\right\}$$
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#### xaktly | Mathematics
Trigonometric functions
### From a black box to a function
So far we've looked at trig functions as ratios of sides, and where those are known, that's easy to calculate. The other way we've used them is as "black boxes" — we put a number into a calculator and we get one back out. Here's a triangle for which we know the lengths of all sides:
\begin{align} sin(\theta) &= \frac{o}{h} = \frac{5}{9.434} = 0.530 \\[5pt] cos(\theta) &= \frac{a}{h} = \frac{8}{9.434} = 0.056 \\[5pt] tan(\theta) &= \frac{o}{a} = \frac{5}{8} = 0.625 \end{align}
Later we'll learn how to recover angles when the side lengths are shown, but let's move on for now.
The other way we've used trigonometry is to enter angle measures (in degrees or radians) into a calculator or computer, and it just gives us the value of the trig function for that angle — magically.
Here's an example of how the trig-function keys will look on your scientific calculator:
It's far from magic, as we will see below, and you'll learn more about that if you study infinite series as part of a calculus course.
### The unit circle – the fundamental basis for trigonometry
#### Units of Angle Measure
In trigonometry and many subfields of math, we use the unit radians for angle measure. The radian arises more or less naturally from the geometry of the circle. There are 2π (yes, 6.283 ..., but we usually just say 2π) radians in a circle, so 180˚ = π radians. The figure on the left shows the most frequently-used radian measures. They are generally multiples of π/6 (30˚) and π/4 (45˚).
You should memorize this unit circle, including the radian measurements. It will grease the wheels for what's ahead. The tool below might help. It reduces the task to simple counting by multiples of 30˚ and 45˚. If you come back to it once in a while, a little more will sink in each time and soon you'll have it.
Note: Unit circle angles begin with zero (degrees or radians) on the far right, and increase in the counterclockwise direction.
There are $2 \pi = 6.28$ radians and 360˚ in a circle, $\pi$ radians and 180˚ in a half circle.
Usually, we don't multiply the $\pi$ when reporting radian measurements. We just say 1.1$\pi$ radians, and so forth.
### Unit circle learning tool
Press the forward arrow as many times as you want to advance the arrow and find the angle, in degrees and radians, on the unit circle.
The circle is "counted around" in 180˚ increments, then in 90˚, 45˚ and 30˚ increments. Go around the circle enough times that you can correctly anticipate the next angle, both in degrees and radians.
Press the back button at any time to reset the circle and start over.
Pro tip: I can't emphasize enough how important it is that you know your way around the unit circle. Now that that's (hopefully) done, we can take a deeper look at the trigonometric functions.
### The special triangles
Two common triangles you should memorize.
These two essential right triangles, their angle measures and the ratios of their side lengths will pop up time and again throughout your studies and work in math and science. Knowing how to jot them down and use them, and from where the measurements came, will be invaluable to you. Study them carefully and memorize their proportions.
### The 45-45-90 triangle
The 45-45-90 triangle is just what it sounds like, a right triangle that has two 45˚ angles, and is thus an isosceles triangle. We'll work with just symbols for the lengths of the sides, like this:
Now the Pythagorean theorem has to hold for this right triangle, so
$$^2 + x^2 = r^2$$
Now combine the like terms on the left,
$$2x^2 = r^2$$
and let's solve for $x$ (you'll see why in a minute), by first dividing both sides by 2.
$$x^2 = \frac{r^2}{2}$$
Now to solve for $x$, take the square root of both sides:
$$x = \sqrt{\frac{r^2}{2}}$$
Taking the square root of the numerator gives r, with $\sqrt{2}$ in the denominator:
$$x = \frac{r}{\sqrt{2}}$$
Finally, just for completeness, not that it really matters mathematically, the denominator of such an equation is usually "rationalized" by multiplying by $\sqrt{2}/\sqrt{2}:$
$$x = \frac{r \sqrt{2}}{2}$$
Now imagine if we let r, the length of the hypotenuse of our triangle, be 1. Then we get this triangle, complete with side measures:
What's great about such a triangle, is that it will give us the lengths of the sides of a 45-45-90 triangle with any hypotenuse length because we have the ratios of the side lengths, $1:\sqrt{2}/2.$
### The 30-60-90 triangle
The 30-60-90 (degrees) right triangle is another frequently-encountered triangle that's worth knowing a lot about.
It isn't quite as easy to find an expression for the side lengths x and y, unless we combine two of these triangles to form an equilateral triangle like this:
Notice here that I've replaced x from the original figure with r/2. We know that the gray vertical line both bisects the top 60˚ angle (by construction) and bisects the base of the triangle. For a proof the the angle bisector of an isosceles triangle also bisects the opposite side, click here.
Now we can set up the Pythagorean theorem for one of the two 30-60-90 triangles shown:
$$r^2 = y^2 + \left( \frac{r}{2} \right)^2$$
Squaring the last term gives
$$r^2 = y^2 + \frac{r^2}{4}$$
Let's solve for y in terms of r by separating terms across the = sign:
$$y^2 = r^2 - \frac{r^2}{4}$$
The difference on the right is easy to calculate; just use 4 as a common denominator:
$$y^2 = \frac{3r^2}{4}$$
Now taking the square root of both sides gives
$$y = r \frac{\sqrt{3}}{2}$$
Now if we take r = 1, we arrive at the prototype 30-60-90 triangle:
So for a 30-60-90 triangle, the shorter leg (recall that the sides that form the right angle of a right triangle are called "legs") has a length that is half the length of the hypotenuse, and the longer leg is $\frac{\sqrt{3}}{2}$ of the hypotenuse.
### Trig ratios in the special triangles
##### $$30^{\circ}, \; \frac{\pi}{6} \text{ rad}$$
\require{cancel} \begin{align} sin(30^{\circ}) &= \frac{1}{2} \\[8pt] cos(30^{\circ}) &= \frac{\sqrt{3}}{2} \\[5pt] tan(30^{\circ}) &= \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\cancel{2}} \frac{\cancel{2}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \end{align}
##### $$45^{\circ}, \; \frac{\pi}{4} \text{ rad}$$
\begin{align} sin(45^{\circ}) &= \frac{\sqrt{2}}{2} \\[8pt] cos(45^{\circ}) &= \frac{\sqrt{2}}{2} \\[5pt] tan(45^{\circ}) &= \frac{\cancel{\frac{\sqrt{2}}{2}}}{\cancel{\frac{\sqrt{2}}{2}}} = 1 \\[5pt] \end{align}
##### $$60^{\circ}, \; \frac{\pi}{3} \text{ rad}$$
\begin{align} sin(60^{\circ}) &= \frac{\sqrt{3}}{2} \\[8pt] cos(60^{\circ}) &= \frac{1}{2} \\[5pt] tan(60^{\circ}) &= \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{\cancel{2}} \frac{\cancel{2}}{1} = \sqrt{3} \\[5pt] \end{align}
### Trigonometric functions
Now we want to use the unit circle to create graphs of the three major trigonometric functions,
\begin{align} f(x) &= sin(x) \\[5pt] f(x) &= cos(x) \\[5pt] f(x) &= tan(x) \end{align}
We'll do that by drawing a series of triangles, both 45-45-90 and 30-60-90, inside of the unit circle, and calculating the trig function values.
Let's start at 30˚ = θ/6. Notice that we can use that triangle to calculate all three trig functions. We'll set those aside as we work around the circle with triangles, and look at all of the values together at the end of the trip.
Now we really should work our way around the circle in increments of 30˚, but I'm going to skip around a bit and hope you'll get the idea. Next is a 135˚ angle, or 45˚ past 90˚, working counterclockwise (as usual) around the unit circle.
One quirk of the unit circle is that the length of the radius (the hypotenuses of all of our triangles) is always 1, but the other triangle sides can have negative or positive values that match with the x,y axis directions. That's how certain sine, cosine or tangent values end up negative.
You might have noticed that the 90˚ (which we passed up) angle doesn't really define a triangle. The key to understanding what happens there is to consider an 89˚ angle, in which the opposite side is very small and the adjacent side is nearly equal to the hypotenuse, making the sine function small (close to zero) and the cosine function close to one. Well, at 90˚, those values are zero and one, respectively, and we make similar arguments at all of the multiples of 90˚ around the circle.
Now let's do a 240˚ = 4θ/3 angle. Notice that because the x-and y-coordinates of the tip of the radius are negative, we treat the sides of our triangle as negative, and that affects the values of the trig functions.
Finally, let's look at an angle in the 4th quadrant, 315˚ = 7θ/4, which is seven increments of 45˚ around the unit circle, and produces a 45-45-90 triangle, just with some "negative" sides.
Notice that all of this would just repeat as we moved our radius vector around the unit circle again and again. Next we'll gather all of this data in a table and determine some patterns in the sine, cosine and tangent functions.
### Trig functions are periodic
Here's a table of trig function values at key angles in one round trip of the unit circle.
The trigonometric functions are repetitive, or cyclic, and we usually refer to them as periodic, meaning that they repeat the same basic pattern predictably.
$\theta$(deg.) $\theta$(rad) sin $(\theta)$ cos $(\theta)$ tan $(\theta)$ 0˚ 0 0 $\frac{\sqrt{0}}{2}$ 1 $\frac{\sqrt{4}}{2}$ 0 30˚ $\frac{\pi}{6}$ $\frac{1}{2}$ $\frac{\sqrt{1}}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{3}$ 45˚ $\frac{\pi}{4}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $1$ 60˚ $\frac{\pi}{3}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\frac{\sqrt{1}}{2}$ $\sqrt{3}$ 90˚ $\frac{\pi}{2}$ $1$ $\frac{\sqrt{4}}{2}$ $0$ $\frac{\sqrt{0}}{2}$ $\infty$ 120˚ $\frac{2\pi}{3}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{2}$ $-\frac{1}{2}$ $-\frac{\sqrt{1}}{2}$ $-\sqrt{3}$ 135˚ $\frac{3\pi}{4}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{2}}{2}$ $-1$ 150˚ $\frac{5\pi}{6}$ $\frac{1}{2}$ $\frac{\sqrt{1}}{2}$ $-\frac{\sqrt{3}}{2}$ $-\frac{\sqrt{3}}{2}$ $-\frac{\sqrt{3}}{3}$ 180˚ $\pi$ $0$ $\frac{\sqrt{0}}{2}$ $-1$ $-\frac{\sqrt{4}}{2}$ $0$
Forget about the gray columns for now. You can pick up a bunch of patterns as we walk around the circle and look at sin(θ), cos(θ) and tan(θ). Notice that the tangent function is a different beast, and we'll get to that later. Notice also that both the sine and cosine functions oscillate between ±1 and pass periodically through y = 0.
The gray columns are just re-expressions of the sine and cosine values to the left. Each is just re-stated with a square root in the numerator so that the increasing-decreasing pattern is more obvious.
The signs of the trig functions change according to which quadrant the tip of radius vector is in (UL = upper left, LR = lower right, and so on). This table shows how the sign of each function depends on the quadrant of the angle:
Signs of the Trig. Functions
UR (I)
+
+
+
UL (II)
+
LL (III)
+
LR (IV)
+
### Trig function graphs
Graphs of the sine (black) and cosine (magenta) functions are drawn below as $f(\theta)$ vs.$\theta$, for $\theta$ between $0$ and a littl more than $4\pi$ - twice around the unit circle plus a little more. Notice that these are periodic functions. Every $2\pi$ radians, they repeat, and that goes on infinitely in both positive and negative directions. The negative direction just means traversing the unit circle in the opposite (clockwise) direction.
All of the points in the table above are plotted on this graph. Be sure to think about these graphs for a while.
Notice a few patterns: They oscillate between ±1; they cross or are farthest apart (vertically) at multiples of $\pi/4$ and $\text{sin}(\theta)$ is the same curve as $\text{cos}(\theta)$, but shifted to the right by $\pi/2$ radians or 90˚. Notice also that $\text{sin}(x) = \text{cos}(x)$ when $x$ is $\frac{\pi}{4}$ and then every $\pi$ radians from there on.
These curves are also referred to as "sine waves". We usually don't say "cosine waves" because the cosine function is the same as the sine, just shifted by $\pi/2$. It is common to represent many periodic natural phenomena using sine and cosine curves: Waves, the motion of a pendulum, and the motion of a bouncing spring, for example.
### Transformations
In order to model periodic behavior using sines and cosines, we need, as usual, some transformations to translate and scale our function at will. In our typical way (see functions section), the transformations of $f(x) = sin(x)$ are shown on the right.
When we use trig. functions to model real data later, we'll make a slight modification of this prototype equation in order to keep the units straight.
The transformations of the other trig. functions are analogous.
### Horizontal stretching
Move the slider to change $c$ in the function $f(x) = sin(f/c)$ to get an idea of the effect of horizontal stretching on a periodic function. The function is written both as $f(x) = sin(x/c)$ and as $f(x) = sin[(1/c)·x]$.
If this function represented a wave (e.g. a sound, light or water wave) then the horizontal stretching parameter, $c$, would change the wavelength, the domain distance between successive peaks or troughs. The parameter $f = 1/c$ is called the frequency factor.
Notice that for small c, the frequency (number of cycles on the graph) is large, and for small $c$, it's large. Because the sine function is odd, when $c$ is negative, the sine curve is inverted across the x-axis.
(Notice also that the graph disappears when we try to divide by zero!)
### Vertical stretching
Move the slider to see the effect of the vertical-stretching parameter, $A$, on our sine function.
If this function represented a wave like a sound or light wave, the vertical distance between $f(x) = 0$ and the maximum (or minimum) value of the sine function would be the amplitude, which would be proportional to volume (loudness) of sound or brightness (intensity) of light.
Notice that when $A \lt 0$, the function is reflected across the x-axis, as we would expect for any function.
### Vertical and horizontal translation
Vertical translation of a periodic function is performed just like on any other function: Simply add a constant number (k) to the value of the function for each point in the domain. Vertical translation is used in modeling to move the average value of the function up or down. For example, to model a tide that fluctuates by ± 3 feet from an average depth of 10 feet, we'd want a sine or cosine function (turns out it doesn't matter which), that oscillates between 7 and 13 feet. The function would read f(some stuff) + 10.
Horizontal translation is very important when we model waves because it represents a property called phase. In our tide example, we might need to shift our model wave over to the right a bit to get the timing of the tides just right (and we'd also have to adjust the wavelength, which would represent the time between high or low tides). Phase is extremely important in many areas of science like x-ray crystallography (used to study molecular structure) and medical imaging.
### The tangent function
Now let's think about a class of trig functions called the reciprocal trig functions. The tangent is the most prominent of these, but there are three more. We can look at the tangent using SOH-CAH-TOA; we know that
$$sin(\theta) = \frac{o}{h}, \phantom{00} cos(\theta) = \frac{a}{h}, \phantom{00} tan(\theta) = \frac{o}{a}.$$
Now notice that
$$\frac{sin(\theta)}{cos(\theta)} = \frac{\frac{o}{h}}{\frac{a}{h}} = \frac{o}{\cancel{h}} \frac{\cancel{h}}{a} = \frac{o}{a}$$
This means that the tangent function is the ratio of the sine and cosine functions:
$$tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$$
Now because $cos(\theta)$ — the denominator of the tangent expression — can be zero, the graph of the tangent function will have vertical asymptotes, as shown in the graph. The tangent of odd multiples of $\pi/2$ is infinite because $cos(\pi/2)$ (and odd multiples of $\pi/2$) is zero.
This has an interesting consequence if you are a rock climber or a lineperson (person who strings cable, e.g. electric wires, from pole to pole. You'll find an explanation here.
There are three other trigonometric functions left to explore, and like the tangent, all are ratios of other trig. functions, so we can expect asymptotic behavior there, too.
$$tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$$
### All six trigonometric functions
Finally, let's define the three other trig. functions. The reciprocal of the sine function (not to be confused with the inverse) is called the cosecant (csc) function. The reciprocal of the cosine is (paradoxically, I know) the secant (sec) and the reciprocal of the tangent is the cotangent (cot). csc(θ), sec(θ) and cot(θ) are defined below.
#### Basic
\begin{align} \text{sine} &\phantom{000} sin(\theta) = \frac{o}{h} \\[5pt] \text{cosine} &\phantom{000} cos(\theta) = \frac{a}{h} \\[5pt] \text{tangent} &\phantom{000} tan(\theta) = \frac{o}{a} \tag{*} \end{align}
#### Reciprocal
\begin{align} \text{cosecant} &\phantom{000} \text{csc} \phantom{00} csc(\theta) = \frac{1}{sin(\theta)} = \frac{h}{o} \\[5pt] \text{secant} &\phantom{000} \text{sec} \phantom{00} sec(\theta) = \frac{1}{cos(\theta)} = \frac{h}{a} \\[5pt] \text{cotangent} &\phantom{000} \text{cot} \phantom{00} cot(\theta) = \frac{cos(\theta)}{sim(\theta)} = \frac{a}{o} \end{align}
(*) Strictly speaking, the tangent is a reciprocal trig function, too, but we'll consider it here to be one of the "big three" most-frequently-used trig functions.
#### Pro tip: Remembering sec, csc and tan
It should be easier to remember which of the main trig. functions are connected to sec, csc and tan, but due to an accident of history, it's not. Sine should go with secant, and cosine should go with cosecant. Unfortunately, those are reversed, and you'll have to remember that. Tangent goes with cotangent, though!
### Graphs of all six trig functions
Several cycles of each of the six trig functions are plotted below. Note that the sin(x) and cos(x) graphs are vertically expanded by a factor of ten compared to the other graphs. This means that the sine graph would fit in between the U-shaped graphs (they are not parabolas!) in the csc(x) graph, same for the cos(x) and sec(x) graphs. Except for sine and cosine,
### Example 1
Determine the values (to the thousandths place) of all six trigonometric functions of the angle $\theta$.
Solution: First find the length of the missing side using the Pythagorean theorem:
\begin{align} o &= \sqrt{21^2 - 17^2} \\[5pt] &= 12.329 \text{ cm} \end{align}
Now that all of the sides are known, we can calculate all of the ratios:
\begin{align} sin(\theta) &= \frac{12.329}{} = 0.587 \\[5pt] cos(\theta) &= \frac{17}{21} = 0.810 \end{align}
\begin{align} tan(\theta) &= \frac{12.329}{17} = 0.725 \\[5pt] csc(\theta) &= \frac{21}{12.329} = 1.703 \\[5pt] sec(\theta) &= \frac{21}{17} = 1.235 \\[5pt] cot(\theta) &= \frac{17}{12.329} = 1.379 \\[5pt] \end{align}
There is much more to trigonometry than can fit into one section like this. You'll need to know how to use inverse trig. functions, how to relate trig functions to one another (analytic trig), and how to use trigonometry on non-right triangles. Here are links to other trigonometry-related pages:
### Practice problems
For problems 1 and 2, calculate the measure of any missing angles and the lengths of any missing sides of the triangle.
1.
Solution
First, the length of the hypotenuse is
$$h = \sqrt{63^2 + 14^2} = \sqrt{4165} = 64.537$$
Now the trig functions are:
\begin{align} sin(\theta) &= \frac{14}{64.537} = 0.217 \\[5pt] cos(\theta) &= \frac{63}{64.537} = 0.976 \\[5pt] tan(\theta) &= \frac{14}{63} = 0.222 \\[5pt] csc(\theta) &= \frac{64.537}{14} = 4.61 \\[5pt] sec(\theta) &= \frac{64.537}{63} = 1.024 \\[5pt] cot(\theta) &= \frac{63}{14} = 4.50 \end{align}
2.
Solution
First, the length of the adjacent side is
$$h = \sqrt{21^2 + 8.5^2} = \sqrt{368.75} = 19.203$$
Now the trig functions are:
\begin{align} sin(\theta) &= \frac{8.5}{21} = 0.405 \\[5pt] cos(\theta) &= \frac{19.203}{21} = 0.914 \\[5pt] tan(\theta) &= \frac{8.5}{19.203} = 0.443 \\[5pt] csc(\theta) &= \frac{21}{8.5} = 2.47 \\[5pt] sec(\theta) &= \frac{21}{19.203} = 1.094 \\[5pt] cot(\theta) &= \frac{19.203}{14} = 2.259 \end{align}
1. The cosine of an angle of a right triangle is $\frac{3}{11}$. Calculate the values of the other five trigonometric functions of this angle, and sketch the triangle.
Solution
Here's a diagram of the triangle, including the length of the missing side:
The missing side is $o = \sqrt{11^2 - 3^2} = 10.583$.
Now the values of the six trig functions are:
\begin{align} sin(x) &= \frac{10.583}{11} = 0.962 \\[5pt] cos(x) &= \frac{3}{11} = 0.273 \\[5pt] tan(x) &= \frac{10.583}{3} = 3.528 \\[5pt] csc(x) &= \frac{11}{10.583} = 1.039 \\[5pt] sec(x) &= \frac{11}{3} = 3.667 \\[5pt] cot(x) &= \frac{3}{10.583} = 0.283 \end{align}
2. The tangent of an angle of a right triangle is $\frac{7}{5}$. Calculate the values of the other five trigonometric functions of this angle and sketch the triangle.
Solution
Here's a diagram of the triangle, including the length of the missing side:
The missing side is $o = \sqrt{11^2 - 3^2} = 10.583$.
Now the values of the six trig functions are:
\begin{align} sin(x) &= \frac{10.583}{11} = 0.962 \\[5pt] cos(x) &= \frac{3}{11} = 0.273 \\[5pt] tan(x) &= \frac{10.583}{3} = 3.528 \\[5pt] csc(x) &= \frac{11}{10.583} = 1.039 \\[5pt] sec(x) &= \frac{11}{3} = 3.667 \\[5pt] cot(x) &= \frac{3}{10.583} = 0.283 \end{align}
3. A certain point on the unit circle is $\left(\frac{7}{25}, \; y \right)$ with $y \gt 0$. Determine the exact value of $y$.
Solution
Here's a diagram of the unit circle with our triangle roughly placed in it:
Now it's clear that our task is just to find the missing side of the triangle, the $y$ coordinate of the point on the circle. That is
\begin{align} y &= \sqrt{1^2 - \left( \frac{7}{25} \right)^2} \\[5pt] &= \sqrt{\frac{625-49}{625}} \\[5pt] &= \sqrt{\frac{576}{625}} \\[5pt] &= \frac{24}{25} \end{align}
4. A certain point on the unit circle is $\left(\frac{15}{17}, \; y \right)$ with $y \lt 0$. Determine the exact value of $y$.
Solution
Here's a diagram of the unit circle with our triangle roughly placed in it:
Now it's clear that our task is just to find the missing side of the triangle, the $y$ coordinate of the point on the circle. That is
\begin{align} y &= -\sqrt{1^2 - \left( \frac{15}{17} \right)^2} \\[5pt] &= -\sqrt{\frac{289-225}{289}} \\[5pt] &= -\sqrt{\frac{64}{289}} \\[5pt] &= -\frac{8}{17} \end{align}
5. Sketch a graph of $f(x) = sin(x) + cos(x)$ over the domain $[0, \, 2\pi]$.
Solution
Let's make a quick table of sums of sin(x) and cos(x) at a few select points:
$\theta$ $sin(\theta)$ $cos(\theta)$ $s(\theta) + c(\theta)$ $0^{\circ}$ $0$ $1$ $1$ $45^{\circ}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $\sqrt{2}$ $90^{\circ}$ $1$ $0$ $1$ $135^{\circ}$ $\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{2}}{2}$ $0$ $180^{\circ}$ $0$ $-1$ $-1$ $225^{\circ}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{2}}{2}$ $-\sqrt{2}$ $270^{\circ}$ $\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{2}}{2}$ $0$ $305^{\circ}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{2}}{2}$ $-\sqrt{2}$
Here is what the graph looks like:
It's still a periodic function with a period of $2 \pi$ radians or 360˚, but it has been shifted by -45˚.
6. Sketch a graph of $g(x) = sin^2(x) + cos^2(x)$. Note that $sin^(x)$ is just a commonly-used shorthand notation for $[sin(x)]^2$, just the square of the result of $sin(x)$.
Solution
Let's make a quick table of sums of sin2(x) and cos2(x) at a few select points:
$\theta$ $sin^2(\theta)$ $cos^(\theta)$ $s^2(\theta) + c^2(\theta)$ $0^{\circ}$ $0$ $1$ $1$ $45^{\circ}$ $\frac{1}{2}$ $\frac{1}{2}$ $1$ $90^{\circ}$ $1$ $0$ $1$ $135^{\circ}$ $\frac{1}{2}$ $\frac{1}{2}$ $1$ $180^{\circ}$ $0$ $1$ $1$ $225^{\circ}$ $\frac{1}{2}$ $\frac{1}{2}$ $1$ $270^{\circ}$ $\frac{1}{2}$ $\frac{1}{2}$ $1$ $305^{\circ}$ $\frac{1}{2}$ $\frac{1}{2}$ $1$
Notice that the sum $sin^2(\theta) + cos^2(\theta) = 1$ for all values of $\theta$. We haven't calculated any values in between, but it's true. This is a relationship that we'll use a lot going into analytic trigonometry and it's called the Pythagorean Identity:
$$sin^2(\theta) + cos^2(\theta) = 1$$
For any angle, $\theta$.
### Video examples
#### 1. Counting around the unit circle
You should know how to label the most-frequently-used angles around the unit circle (circle of radius r = 1), both in degrees and radians. It's not that difficult if you just think of it as counting around the circle in different increments.
#### Triangles
Back up and refresh your memory about the properties of triangles, key to understanding trigonometry.
#### Trigonometric functions
Learn about the properties and graphs of the trigonometric functions, sin(θ), cos(θ), tan(θ) and three others.
X
### mnemonic
A mnemonic (nee·mon'·ick) is a word or phrase designed to help a person remember something. An example would be the pseudo-word "ROYGBIV" or the phrase "Rogers of York Gave Battle in Vain." Both are designed to help us remember the colors of the visible spectrum: red, orange, yellow, green, blue, indigo & violet.
X
#### The Greek alphabet
alpha Α α beta Β β gamma Γ γ delta Δ δ epsilon Ε ε zeta Ζ ζ eta Η η theta Θ θ iota Ι ι kappa Κ κ lambda Λ λ mu Μ μ nu Ν ν xi Ξ ξ omicron Ο ο pi Π π rho Ρ ρ sigma Σ σ tau Τ τ upsilon Υ υ phi Φ φ chi Χ χ psi Ψ ψ omega Ω ω
xaktly.com by Dr. Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. © 2012-2024, Jeff Cruzan. All text and images on this website not specifically attributed to another source were created by me and I reserve all rights as to their use. Any opinions expressed on this website are entirely mine, and do not necessarily reflect the views of any of my employers. Please feel free to send any questions or comments to [email protected].
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teaching resource
# Number Line Multiplication – Interactive Picture Reveal
• Updated: 20 Nov 2023
Engage your students with this interactive multiplication game while learning how to multiply on a number line.
• Pages: 1 Page
• Years: 3 - 4
Tag #TeachStarter on Instagram for a chance to be featured!
teaching resource
# Number Line Multiplication – Interactive Picture Reveal
• Updated: 20 Nov 2023
Engage your students with this interactive multiplication game while learning how to multiply on a number line.
• Pages: 1 Page
• Years: 3 - 4
Engage your students with this interactive multiplication game while learning how to multiply on a number line.
## How to Practise Multiplication With Number Lines
If you are teaching beginning multiplication concepts to your students, chances are you’ve taught them about arrays, equal groups, skip counting and repeated addition. Another common multiplication strategy is to use a number line to represent multiplication problems. So how is this done, exactly?
To represent a multiplication problem on a number line, guide your students to think of a problem, such as 6 x 4, as ‘6 hops of 4.’ This will help them know how many jumps to make on the number line. Starting at 0, students can then model the equal-sized jumps until they reach the product of 24. It is important to point out to your students that the number lines can be scaled, meaning they all may not increase by 1. Some number lines may be scaled by 2, 3, 5 or 10.
## Add This Digital Activity to Your Collection of Maths Games for Year 3!
Are you looking for a self-checking activity to help your students understand the relationship between number lines and multiplication? If so, you have come to the right place! Teach Starter has developed an exciting interactive mystery reveal game to help your students determine multiplication facts represented by a number line model.
As your students complete this activity, they will answer 20 questions that test their understanding of number line multiplication. They will also get practise using a variety of scaled number lines to represent problems. The great thing about this activity is that students will get instant feedback on whether or not they answered each question correctly.
## How to Get Your Interactive Multiplication Practice Problems
If you are ready for your students to dive into this engaging activity, head on over to the green download button! Here, you will be able to download the interactive Google Slides or PowerPoint file. If selecting the Google Slides option, please note that you will first be prompted to make a copy of the resource to your personal drive before accessing it.
When assigning this activity to your students, guide them to open the file in ‘Presentation’ mode. Students will then answer each of the questions and click on the correct answer. If the student is right, they will progress to the next question. If they are wrong, they will be sent back to try again.
As students complete each of the problems, a piece of the puzzle grid will disappear, eventually revealing a hidden picture underneath.
This resource was created by Cassandra Friesen, a Teach Starter Collaborator.
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# 13.6 Binomial theorem (Page 3/6)
Page 3 / 6
## Key equations
Binomial Theorem ${\left(x+y\right)}^{n}=\sum _{k-0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}$ $\left(r+1\right)th\text{\hspace{0.17em}}$ term of a binomial expansion $\left(\begin{array}{c}n\\ r\end{array}\right){x}^{n-r}{y}^{r}$
## Key concepts
• $\left(\begin{array}{c}n\\ r\end{array}\right)\text{\hspace{0.17em}}$ is called a binomial coefficient and is equal to $C\left(n,r\right).\text{\hspace{0.17em}}$ See [link] .
• The Binomial Theorem allows us to expand binomials without multiplying. See [link] .
• We can find a given term of a binomial expansion without fully expanding the binomial. See [link] .
## Verbal
What is a binomial coefficient, and how it is calculated?
A binomial coefficient is an alternative way of denoting the combination $\text{\hspace{0.17em}}C\left(n,r\right).\text{\hspace{0.17em}}$ It is defined as $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ r\end{array}\right)=\text{\hspace{0.17em}}C\left(n,r\right)\text{\hspace{0.17em}}=\frac{n!}{r!\left(n-r\right)!}.$
What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?
What is the Binomial Theorem and what is its use?
The Binomial Theorem is defined as $\text{\hspace{0.17em}}{\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}\text{\hspace{0.17em}}$ and can be used to expand any binomial.
When is it an advantage to use the Binomial Theorem? Explain.
## Algebraic
For the following exercises, evaluate the binomial coefficient.
$\left(\begin{array}{c}6\\ 2\end{array}\right)$
15
$\left(\begin{array}{c}5\\ 3\end{array}\right)$
$\left(\begin{array}{c}7\\ 4\end{array}\right)$
35
$\left(\begin{array}{c}9\\ 7\end{array}\right)$
$\left(\begin{array}{c}10\\ 9\end{array}\right)$
10
$\left(\begin{array}{c}25\\ 11\end{array}\right)$
$\left(\begin{array}{c}17\\ 6\end{array}\right)$
12,376
$\left(\begin{array}{c}200\\ 199\end{array}\right)$
For the following exercises, use the Binomial Theorem to expand each binomial.
${\left(4a-b\right)}^{3}$
$64{a}^{3}-48{a}^{2}b+12a{b}^{2}-{b}^{3}$
${\left(5a+2\right)}^{3}$
${\left(3a+2b\right)}^{3}$
$27{a}^{3}+54{a}^{2}b+36a{b}^{2}+8{b}^{3}$
${\left(2x+3y\right)}^{4}$
${\left(4x+2y\right)}^{5}$
$1024{x}^{5}+2560{x}^{4}y+2560{x}^{3}{y}^{2}+1280{x}^{2}{y}^{3}+320x{y}^{4}+32{y}^{5}$
${\left(3x-2y\right)}^{4}$
${\left(4x-3y\right)}^{5}$
$1024{x}^{5}-3840{x}^{4}y+5760{x}^{3}{y}^{2}-4320{x}^{2}{y}^{3}+1620x{y}^{4}-243{y}^{5}$
${\left(\frac{1}{x}+3y\right)}^{5}$
${\left({x}^{-1}+2{y}^{-1}\right)}^{4}$
$\frac{1}{{x}^{4}}+\frac{8}{{x}^{3}y}+\frac{24}{{x}^{2}{y}^{2}}+\frac{32}{x{y}^{3}}+\frac{16}{{y}^{4}}$
${\left(\sqrt{x}-\sqrt{y}\right)}^{5}$
For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.
${\left(a+b\right)}^{17}$
${a}^{17}+17{a}^{16}b+136{a}^{15}{b}^{2}$
${\left(x-1\right)}^{18}$
${\left(a-2b\right)}^{15}$
${a}^{15}-30{a}^{14}b+420{a}^{13}{b}^{2}$
${\left(x-2y\right)}^{8}$
${\left(3a+b\right)}^{20}$
$3,486,784,401{a}^{20}+23,245,229,340{a}^{19}b+73,609,892,910{a}^{18}{b}^{2}$
${\left(2a+4b\right)}^{7}$
${\left({x}^{3}-\sqrt{y}\right)}^{8}$
${x}^{24}-8{x}^{21}\sqrt{y}+28{x}^{18}y$
For the following exercises, find the indicated term of each binomial without fully expanding the binomial.
The fourth term of $\text{\hspace{0.17em}}{\left(2x-3y\right)}^{4}$
The fourth term of $\text{\hspace{0.17em}}{\left(3x-2y\right)}^{5}$
$-720{x}^{2}{y}^{3}$
The third term of $\text{\hspace{0.17em}}{\left(6x-3y\right)}^{7}$
The eighth term of $\text{\hspace{0.17em}}{\left(7+5y\right)}^{14}$
$220,812,466,875,000{y}^{7}$
The seventh term of $\text{\hspace{0.17em}}{\left(a+b\right)}^{11}$
The fifth term of $\text{\hspace{0.17em}}{\left(x-y\right)}^{7}$
$35{x}^{3}{y}^{4}$
The tenth term of $\text{\hspace{0.17em}}{\left(x-1\right)}^{12}$
The ninth term of $\text{\hspace{0.17em}}{\left(a-3{b}^{2}\right)}^{11}$
$1,082,565{a}^{3}{b}^{16}$
The fourth term of $\text{\hspace{0.17em}}{\left({x}^{3}-\frac{1}{2}\right)}^{10}$
The eighth term of $\text{\hspace{0.17em}}{\left(\frac{y}{2}+\frac{2}{x}\right)}^{9}$
$\frac{1152{y}^{2}}{{x}^{7}}$
## Graphical
For the following exercises, use the Binomial Theorem to expand the binomial $f\left(x\right)={\left(x+3\right)}^{4}.$ Then find and graph each indicated sum on one set of axes.
Find and graph $\text{\hspace{0.17em}}{f}_{1}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{1}\left(x\right)\text{\hspace{0.17em}}$ is the first term of the expansion.
Find and graph $\text{\hspace{0.17em}}{f}_{2}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{2}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first two terms of the expansion.
${f}_{2}\left(x\right)={x}^{4}+12{x}^{3}$
Find and graph $\text{\hspace{0.17em}}{f}_{3}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{3}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first three terms of the expansion.
Find and graph $\text{\hspace{0.17em}}{f}_{4}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{4}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first four terms of the expansion.
${f}_{4}\left(x\right)={x}^{4}+12{x}^{3}+54{x}^{2}+108x$
Find and graph $\text{\hspace{0.17em}}{f}_{5}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{5}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first five terms of the expansion.
## Extensions
In the expansion of $\text{\hspace{0.17em}}{\left(5x+3y\right)}^{n},\text{\hspace{0.17em}}$ each term has the form successively takes on the value $\text{\hspace{0.17em}}0,1,2,\text{\hspace{0.17em}}...,\text{\hspace{0.17em}}n.$ If $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}7\\ 2\end{array}\right),\text{\hspace{0.17em}}$ what is the corresponding term?
$590,625{x}^{5}{y}^{2}$
In the expansion of $\text{\hspace{0.17em}}{\left(a+b\right)}^{n},\text{\hspace{0.17em}}$ the coefficient of $\text{\hspace{0.17em}}{a}^{n-k}{b}^{k}\text{\hspace{0.17em}}$ is the same as the coefficient of which other term?
Consider the expansion of $\text{\hspace{0.17em}}{\left(x+b\right)}^{40}.\text{\hspace{0.17em}}$ What is the exponent of $b$ in the $k\text{th}$ term?
$k-1$
Find $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{c}n\\ k\end{array}\right)\text{\hspace{0.17em}}$ and write the answer as a binomial coefficient in the form $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k\end{array}\right).\text{\hspace{0.17em}}$ Prove it. Hint: Use the fact that, for any integer $\text{\hspace{0.17em}}p,\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}p\ge 1,\text{\hspace{0.17em}}p!=p\left(p-1\right)!\text{.}$
$\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{l}n\\ k\end{array}\right)=\left(\begin{array}{c}n+1\\ k\end{array}\right);\text{\hspace{0.17em}}$ Proof:
$\begin{array}{}\\ \\ \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{l}n\\ k\end{array}\right)\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k-1\right)!\left(n-\left(k-1\right)\right)!}\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k-1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!}{\left(n-k+1\right)k!\left(n-k\right)!}+\frac{kn!}{k\left(k-1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!+kn!}{k!\left(n-k+1\right)!}\\ =\frac{\left(n+1\right)n!}{k!\left(\left(n+1\right)-k\right)!}\\ =\frac{\left(n+1\right)!}{k!\left(\left(n+1\right)-k\right)!}\\ =\left(\begin{array}{c}n+1\\ k\end{array}\right)\end{array}$
Which expression cannot be expanded using the Binomial Theorem? Explain.
• $\left({x}^{2}-2x+1\right)$
• ${\left(\sqrt{a}+4\sqrt{a}-5\right)}^{8}$
• ${\left({x}^{3}+2{y}^{2}-z\right)}^{5}$
• ${\left(3{x}^{2}-\sqrt{2{y}^{3}}\right)}^{12}$
The expression $\text{\hspace{0.17em}}{\left({x}^{3}+2{y}^{2}-z\right)}^{5}\text{\hspace{0.17em}}$ cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.
By the definition, is such that 0!=1.why?
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
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| null |
Atherosclerosis is a hardening of your arteries due to gradual plaque buildup. Risk factors include high cholesterol, high blood pressure, diabetes, tobacco use, obesity, lack of exercise and a diet high in saturated fat. Atherosclerosis develops over time and may not show symptoms until you have complications like a heart attack or stroke.
Atherosclerosis is the gradual buildup of plaque in the walls of your arteries. Arteries are blood vessels that carry oxygen-rich blood to organs and tissues throughout your body. Plaque (atheroma) is a sticky substance made of fat, cholesterol, calcium and other substances.
As plaque builds up, your artery wall grows thicker and harder. This “hardening of the arteries” is usually a silent process in the early stages. You may not notice symptoms for a long time. But eventually, as the plaque grows, the opening (lumen) of your artery narrows, leaving less room for blood to flow. This means less blood can reach your organs and tissues. Plus, the constant force of blood flow can lead to plaque erosion or rupture, causing a blood clot to form.
A narrowed artery is like a highway reduced to one lane. But a blood clot is like a barricade in the middle of the road. It blocks blood flow to certain organs or tissue the artery normally feeds. The effects on your body depend on where the blood clot forms. For example, blockages in a coronary artery deprive your heart of oxygen-rich blood, leading to a heart attack.
But there’s a reason for hope. You can lower your risk for atherosclerosis, or slow its progression, by making lifestyle changes and managing underlying conditions. Research shows some treatments can reduce the size of plaque in your arteries (plaque regression) or change its chemical makeup, so it’s less likely to rupture.
That’s why visiting a healthcare provider for yearly checkups is important. They’ll evaluate your risk for atherosclerosis and explain what you can do to lower it.
Atherosclerosis interferes with the normal workings of your cardiovascular system. It can limit or block blood flow to various parts of your body, including your heart and brain. Possible complications of reduced blood flow include:
Atherosclerosis can also weaken your artery walls, leading to the formation of aneurysms.
Early diagnosis and treatment of atherosclerosis can help you avoid or delay complications.
Atherosclerosis is very common. The complications of plaque buildup (including heart attacks and strokes) are the leading cause of death worldwide.
About half of people age 45 to 84 have atherosclerosis but aren’t aware of it, according to the U.S. National Institutes of Health.
If you have warning signs of atherosclerosis, tell a healthcare provider. Early treatment can lower your risk of life-threatening complications.
Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services. Policy
Atherosclerosis often doesn’t cause symptoms until an artery is very narrow or blocked. Many people don’t know they have plaque buildup until a medical emergency, like a heart attack or stroke, occurs.
You may notice symptoms if your artery is more than 70% blocked. Listed below are common complications and possible associated symptoms:
Coronary artery disease
Peripheral artery disease (PAD)
Renal artery stenosis
Stroke or transient ischemic attack (TIA)
Carotid artery disease
Call 911 or your local emergency number right away if you or someone near you has symptoms of a heart attack, stroke or TIA. These are medical emergencies that require immediate care.
Damage to your artery’s inner lining (endothelium) causes atherosclerosis to begin. The damage usually occurs slowly and over time.
The stages of atherosclerosis happen over many years and include:
There are many risk factors for atherosclerosis. Non-modifiable risk factors are those you can’t change. You may be able to reduce modifiable risk factors, including some medical conditions and lifestyle factors, in some cases.
It’s important to note that some risk factors vary based on your sex assigned at birth. For example, people assigned male at birth (AMAB) face a higher risk of atherosclerosis at a younger age than people assigned female at birth (AFAB).
Non-modifiable risk factors
Talk to your provider about your risks and what you can do to lower them.
To diagnose atherosclerosis or calculate your risk for developing it, a healthcare provider will:
Your healthcare provider may order additional tests to diagnose atherosclerosis and plan treatment. These tests include:
If you have atherosclerosis, your healthcare provider may recommend you see a specialist, such as a:
Atherosclerosis treatment includes one or more of the following:
Your healthcare provider will develop a plan based on your needs. Common treatment goals include:
Lifestyle changes may lower your risk of complications. Your provider will create a plan specific to your needs. General tips include:
Medications target risk factors for plaque buildup and may help slow the progression of atherosclerosis. Your provider may prescribe medications to:
It’s important to take all of your medications as prescribed. Always check with your provider before making any changes to your medication schedule.
Various minimally invasive procedures and complex surgeries can help people with severe blockages or a high risk of complications. Common treatment options include:
You may not be able to prevent atherosclerosis. But you can reduce your risk and lessen the effects of the disease. Here are some steps you can take:
Early diagnosis and treatment can help people with atherosclerosis live healthy, active lives. But the disease can cause medical emergencies and even be fatal. That’s why knowing your risks and working with your healthcare provider to lower them is important.
It’s essential to work closely with your healthcare provider. They’ll keep a close eye on your condition and tell you how often you should come in for appointments. Go to all of your appointments and follow-ups, and be an active partner in your care. Tell your provider right away about any new or changing symptoms.
Also, take care of your mental health. It’s normal to feel anxious about what the future could bring. You may also feel overwhelmed by the need to make lifestyle changes. But those feelings shouldn’t prevent you from enjoying life. Some tips for managing your thoughts and worries include:
A note from Cleveland Clinic
Atherosclerosis is a common condition. So, remember that if you have atherosclerosis, you’re not alone. Many other people are in your shoes. Your healthcare provider is ready to help you manage your condition so you can live a long and healthy life.
Last reviewed by a Cleveland Clinic medical professional on 03/06/2023.
Learn more about our editorial process.
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1. Linear transformations
If T:P2 -> M22 is a linear transform such that:
T(1) = |1 0|
.........|0 1|
T(1+x) = |1 1|
.............|0 1|
T(1+x+x^2) = |0 -1|
....................|1 0|
Find T(5-3x+2x^2)
There are a couple ways I know of solving this:
One is to use the fact that the three polynomials are a basis of P2:
Thus 5-3x+2x^2 = c_1*(1) + c_2*(1+x) + c_3*(1+x+x^2)
I need to determine c_1, c_2, and c_3.
So, I think:
T(5-3x+2x^2) = c_1*T(1) + c_2*T(1+x) + c_3*T(1+x+x^2)
However, I am having problems equating c1 c2 and c3.
I came up with c_1 =5, c_2=-3, and c_3=2
but I do not recall I equated coefficients on the two sides of the equation properly.
That is:
The coefficient of x on the LHS is -3, and the RHS = c_2 + c_3 thus:
c_2 + c_3 = -3
Once we know those values (c_1, c_2, and c_3):
Then I "think" we need to sub them into:
T(c_1*v_1 + c_2*v_2) = c_1*T(v_1) + c_2*T(v_2)
Then if we can get the values of c_1 c_2 and c_3...
for the vectors v_1 = 1, v_2 = 1+x, and v_3 = 1+x+x^2.
Therefore:
T(c_1*v_1+c_2*v_2+c_3*v_3)
= c_1*T(v_1)+c_2*T(v_2)+c_3*T(v_3)
= c_1*|1 0| + c_2*|1 1| + c_3*|0 -1|
..........|0 1|..........|0 1|..........|1 0|
Then we should be able to determine a solution for what is equal to T(5-3x+2x^2).
So.
Apoogies from me to the many experts for showin' so much work on this one!
Bottom line:
I have problems in this one computing c_1 c_2 and c_3.
I am a strong believer in showing "your" work. In general I typically make problems more convoluted this way...but we do "not need" an bunch of people out there posting questions and looking for "easy" answers.
Please advise. I hope someone sees it and can share a correct way.
Note: I mean,
There is "One other way" to solve this but I am no expert:
Use the values of T(1), T(1+x), and T(1+x+x^2) to find the values of T on the std basis of P2...the polynomials 1,x,x^2.
We would still have to evaluate the values of c_1 c_2 and c_3 by expansion similar above then equate coefficients of powers on both sides our equations.
Anyway, if anyone can illustrate either method and it works same answer great.
Much appreciated of you guys!
2. Originally Posted by orendacl
If T:P2 -> M22 is a linear transform such that:
T(1) = |1 0|
.........|0 1|
T(1+x) = |1 1|
.............|0 1|
T(1+x+x^2) = |0 -1|
....................|1 0|
Find T(5-3x+2x^2)
There are a couple ways I know of solving this:
One is to use the fact that the three polynomials are a basis of P2:
Thus 5-3x+2x^2 = c_1*(1) + c_2*(1+x) + c_3*(1+x+x^2)
I need to determine c_1, c_2, and c_3.
So, I think:
T(5-3x+2x^2) = c_1*T(1) + c_2*T(1+x) + c_3*T(1+x+x^2)
However, I am having problems equating c1 c2 and c3.
I came up with c_1 =5, c_2=-3, and c_3=2
but I do not recall I equated coefficients on the two sides of the equation properly.
That is:
The coefficient of x on the LHS is -3, and the RHS = c_2 + c_3 thus:
c_2 + c_3 = -3
Once we know those values (c_1, c_2, and c_3):
Then I "think" we need to sub them into:
T(c_1*v_1 + c_2*v_2) = c_1*T(v_1) + c_2*T(v_2)
Then if we can get the values of c_1 c_2 and c_3...
for the vectors v_1 = 1, v_2 = 1+x, and v_3 = 1+x+x^2.
Therefore:
T(c_1*v_1+c_2*v_2+c_3*v_3)
= c_1*T(v_1)+c_2*T(v_2)+c_3*T(v_3)
= c_1*|1 0| + c_2*|1 1| + c_3*|0 -1|
..........|0 1|..........|0 1|..........|1 0|
Then we should be able to determine a solution for what is equal to T(5-3x+2x^2).
So.
Apoogies from me to the many experts for showin' so much work on this one!
Bottom line:
I have problems in this one computing c_1 c_2 and c_3.
I am a strong believer in showing "your" work. In general I typically make problems more convoluted this way...but we do "not need" an bunch of people out there posting questions and looking for "easy" answers.
Please advise. I hope someone sees it and can share a correct way.
Note: I mean,
There is "One other way" to solve this but I am no expert:
Use the values of T(1), T(1+x), and T(1+x+x^2) to find the values of T on the std basis of P2...the polynomials 1,x,x^2.
We would still have to evaluate the values of c_1 c_2 and c_3 by expansion similar above then equate coefficients of powers on both sides our equations.
Anyway, if anyone can illustrate either method and it works same answer great.
Much appreciated of you guys!
One way to find $\displaystyle c_1, c_2$ and $\displaystyle c_3$ is just what you say: set $\displaystyle 5-3x+2x^2 = c_1*(1) + c_2*(1+x) + c_3*(1+x+x^2)$. Don't worry about "T" yet, just multiply out the right side:
$\displaystyle 5-3x+2x^2 = c_1 + c_2+ c_2x) + c_3+ c_3x+c_3x^2$
$\displaystyle 5-3x+2x^2 = (c_1+ c_2+ c_3)+ (c_2+ c_3)x+ c_2x^2$
Since that is to be true for all x, we can equate the same coefficients on both sides: we must have $\displaystyle 5= c_1+ c_2+ c_3$, $\displaystyle -3= c_2+ c_3$, and $\displaystyle 2= c_3$. Obviously the last equation tells us that $\displaystyle c_3= 2$. Putting that into the second equation, $\displaystyle -3= c_2+ 2$ so $\displaystyle c_2= -5$. Putting those values into the first equation, $\displaystyle 5= c_1+ (-5)+ (2)$ so $\displaystyle c_1= 8$.
Another way to find those numbers is to use the fact that $\displaystyle 5-3x+2x^2 = c_1*(1) + c_2*(1+x) + c_3*(1+x+x^2)$ is true for all values of x. If we set x= 0 we get $\displaystyle 5+0+0= 5= c_1+ c_2+ c_3$ as before. If we set x= 1, we get $\displaystyle 5-3+2= 4= c_1+ 2c_2+ 7c_3$. If we set x= -1, we get $\displaystyle 5+3+2= 10= c_1+ c_3$
Now you know that $\displaystyle 5-3x+2x^2 = 8(1) - 5(1+x) + 2(1+x+x^2)$ so $\displaystyle T(5-3x+2x^2) = 8T(1) - 5T(1+x) + 2T(1+x+x^2)$
|
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Function Repository Resource:
# PiecewiseD
The derivative of a piecewise function with Indeterminate for points or regions where the function is not defined
Contributed by: Dennis M Schneider
ResourceFunction["PiecewiseD"][f,x] returns the derivative of a piecewise function returning the value Indeterminate for points or regions where the function is not defined. ResourceFunction["PiecewiseD"][f,x,k] returns the function together with its first k derivatives. ResourceFunction["PiecewiseD"][f,{x,k}] returns the kth derivative.
## Examples
### Basic Examples (2)
Compute the derivative of a piecewise function:
In[1]:=
Out[2]=
Compute the derivatives of a function whose domain is not an interval:
In[3]:=
Out[4]=
Compute just the third derivative:
In[5]:=
Out[5]=
Plot the function together with its first three derivatives:
In[6]:=
Out[6]=
### Scope (4)
Find and plot the first- and second-order derivatives. The function and its first-order derivative are continuous at x=0, but not the second-order derivative:
In[7]:=
Out[8]=
Check that the first derivative is continuous:
In[9]:=
Out[9]=
Check that the second derivative is not continuous:
In[10]:=
Out[10]=
Plot the results:
In[11]:=
Out[11]=
### Properties and Relations (1)
Show the difference between PiecewiseD and D:
In[12]:=
Out[12]=
### Applications (5)
The following function has a removable discontinuity at x=3 and an infinite discontinuity at x=4:
In[13]:=
Out[14]=
Extend the definition at x=3 to make the extended function continuous there:
In[15]:=
Out[15]=
In[16]:=
Out[17]=
The extended function is actually differentiable at x=3:
In[18]:=
Out[18]=
The resource function EnhancedPlot produces a correct plot:
In[19]:=
Out[19]=
The function g is differentiable at x=0 and PiecewiseD returns the correct value, 1. The function D, however, returns the value 0 for the derivative at x=0:
In[20]:=
Out[22]=
In[23]:=
Out[23]=
However, the derivative is not continuous:
In[24]:=
Out[24]=
This function is differentiable at x=0 and its derivative is continuous there:
In[25]:=
Out[27]=
In[28]:=
Out[28]=
Plot the result using the resource function EnhancedPlot:
In[29]:=
Out[29]=
A function with a singularity at x=-1 and x=1; PiecewiseD returns the correct result. Note that if this expression is simplified, the singularity at x=1 will be lost:
In[30]:=
Out[30]=
The resource function EnhancedPlot is able to produce a correct plot:
In[31]:=
Out[31]=
Extend the function so that it becomes continuous at -1 and 1:
In[32]:=
Out[32]=
In[33]:=
The first and second derivatives are continuous at ±1:
In[34]:=
Out[34]=
In[35]:=
Out[35]=
Plot the extended function and its first two derivatives:
In[36]:=
Out[36]=
A classic example of a nonzero infinitely differentiable function all of whose derivatives at x=0 are 0 and hence all of whose Taylor polynomials based at 0 are the zero polynomial:
In[37]:=
Out[39]=
Illustrate with ResourceFunction["EnhancedPlot"]:
In[40]:=
Out[40]=
## Publisher
Dennis M Schneider
## Version History
• 2.0.1 – 16 August 2022
• 2.0.0 – 09 August 2022
• 1.0.0 – 23 September 2020
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crawl-data/CC-MAIN-2024-38/segments/1725700650826.4/warc/CC-MAIN-20240907095856-20240907125856-00176.warc.gz
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Conditioned and unconditioned responses are behaviors that result from specific stimuli. An unconditioned response is behavior that occurs naturally due to a given stimulus. However, a stimulus prompts a conditioned response only when someone has come to associate that stimulus with another. For example, when a person yelps upon being bitten by an insect, the yelp is an unconditioned response. After hearing a buzzing every time one is bitten, one might begin to yelp every time one notices the sound: this is a conditioned response because it occurs after one learns to associate the buzz with an insect bite (it does not occur spontaneously). Psychologists distinguish between conditioned and unconditioned responses to explain classical conditioning, a kind of learning.
Get help on Psychology with Chegg Study
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In social sciences there are many key concepts and terms that are crucial for students to know and understand. Often it can be hard to determine what the most important social sciences concepts and terms are, and even once you’ve identified them you still need to understand what they mean. To help you learn and understand key social sciences terms and concepts, we’ve identified some of the most important ones and provided detailed definitions for them, written and compiled by Chegg experts.
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# The given table shows the number of subscribers, three newspapers A, B and C have in 4 regions of the country, W, X, Y and Z. The country is divided into these 4 regions only. Each subscriber subscribes to only 1 newspaper. After newspaper B reduces its price by 20%, 20% of subscribers of newspaper A and 40% of subscribers of newspaper C switch to newspaper B, then revenue of newspaper B Increases by ______. Newspaper/City W X Y Z A 20 30 40 10 B 20 40 30 10 C 40 20 30 10
This question was previously asked in
UPSSSC Mandi Inspector Official Paper 1 (Held On : 30 May 2019 Shift 1)
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1. 32%
2. 40%
3. 60%
4. 28%
Option 4 : 28%
## Detailed Solution
Calculation:
Let the initial price of newspaper B be 5x.
Initial revenue = (20 + 40 + 30 + 10) × 5x
⇒ Initial revenue = 500x
Final subscriber of B = 100 + (20% of 100 + 40% of 100)
⇒ Final subscriber of B = 100 + (20 + 40)
⇒ Final subscriber of B = 160
Final revenue = 160 × 4x
⇒ Final revenue = 640x
Difference = 640x - 500x
⇒ Difference = 140x
Percentage increase = (140x/500x) × 100%
⇒ Percentage increase = 28%
∴ The increase in the revenue of B is 28%.
|
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Scientific name: Thymelicus lineola
The Essex Skipper is a small butterfly, widespread in England and Wales. It has a darting flight and holds the forewings angled above the hind wings.
The bright orange-brown upperwings have a black border and faint black lines. The underwings are plain and paler in colour.
The male can be distinguied from the female by the thin black line of scent scales through the centre of the forewing, parallel to the leading edge. It actively patrols its territories, while the female is conspicuous. They both nectar from flowers such as Red Clover and Thistles and rest and bask on grass stems.
The Essex Skipper is easily confused with the Small Skipper, which is very similar but lacks the glossy black-tipped antenna (best viewed head on). It also has a longer scent mark, angled to the edge of the forewing and it is slightly more orange that the Essex Skipper. Because of the similarities, the Essex Skipper has been overlooked both in terms of recording and ecological study, and it was the last British resident species to be described (in 1889).
The pale-coloured eggs are laid in a tight leaf sheath. They darken within a few days, turning a creamy yelllow and after three weeks turn white, with the head visible through the shell as a dark spot. The fully-grown caterpillar overwinters in the egg.
The caterpillar emerges in the spring and starts feeding on the foodplant. After a few days it forms a tube, pulling the edges of a leaf together with silk threads where it hides when not feeding.
The caterpillar moults five times and, at the end of this stage of its lifecycle is pale green, striped with dark green down the back and yellow along the sides. It forms a tent of leaves and silk threads, where it pupates over three weeks. The chrysalis is yellow-green in colour.
The distribution of the Essex Skipper in Britain has more than doubled in the last few decades, helped, it is believed, by the steep grass embankments of motorways and trunk roads, which provide corridors to new locations for the butterfly. It can be found - despite its name - throughout south-east England as far west as Dorset and Somerset, and north up into the Midlands. It lives in colonies of up to several thousand.
Size and Family
- Family: Skippers
- Size: Small
- Wing Span Range (male to female) - 27-30mm
- Butterfly Conservation priority: Low
- Low European Status: Not threatened
The main species used is Cock’s-foot (Dactylis glomerata), although the butterfly may use several other grasses including Creeping Soft-grass (Holcus mollis), Common Couch (Elytrigia repens), Timothy (Phleum pratense), Meadow Foxtail (Alopecurus pratensis), False Brome (Brachypodium sylvaticum), and Tor-grass (B. pinnatum). It rarely uses Yorkshire-fog (Holcus lanatus), the preferred foodplant of the Small Skipper.
The buttterfly is found in open sunny situations, such as tall, dry grasslands, roadside verges, disused railway lines, woodland rides and acid grasslands, as well as coastal marshes.
- Countries: England and Wales
- Widespread in southern and central England, but not in far south-west
- Distribution Trend Since 1970’s = Britain: +46%
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What is a Seismograph?
What is a Seismograph?
A seismograph is simply a device that can measure the intensity of a “seismic” type of activity such as an earthquake. But aside from the earthquake’s intensity, seismographs can also measure and record the earthquake’s duration and direction. But unknown to many people, seismographs may also be used to record information from other activities such as tidal waves, big explosions, and other occurrences that result in the shaking of the ground.
Many people interchange the word “seismometer” for the “seismograph”. But generally, seismographs refer to older devices and scientific instruments that record and measure seismic activity using a single system. Seismometers meanwhile are used for modern devices which have separate systems and mechanisms for recording and measuring a particular seismic activity.
The basic concept of seismographs or seismometers is that it involves an internal mass or weight since it will also move during an earthquake. Using a system of levers, weights, springs, and electronics in modern versions, seismographs are able to record information on the shaking of the ground. The earliest version of this device was said to originate in China through a device called a “seismoscope”, which basically indicates whether some form of ground motion occurred and perhaps very little information on its intensity. Over the years, this simple device has evolved and has been improved to provide more reliable data on the earth’s seismic activities. Modern devices today employ electronics and are able to detect the earth’s motion using different frequencies. Some seismographs or seismometers today even have the capability of locating the source of an earthquake with great precision technology.
Information taken from seismographs and similar devices has proven very essential to human lives. Though much of the earth’s natural seismic activities cannot be precisely predicted, studies of these events are still a great help in determining its impact to the people. But aside from its use on monitoring seismic activities, seismographs are also used by some government organizations and intelligence agencies to test explosives and artillery.
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The website of the Nobel Prize shows a cat resting in a graphene hammock. Although fictitious, the image captures the excitement around graphene, which, at one atom thick, is the among the thinnest and strongest materials ever produced.
A significant obstacle to realizing graphene's potential lies in creating a surface large enough to support a theoretical sleeping cat. For now, material scientists stitch individual graphene sheets together to create sheets that are large enough to investigate possible applications. Just as sewing patches of fabric together may create weaknesses where individual patches meet, defects can weaken the "grain boundaries" where graphene sheets are stitched together -- at least that is what engineers had thought.
Now, engineers at Brown University and the University of Texas-Austin have discovered that the grain boundaries do not compromise the material's strength. The grain boundaries are so strong, in fact, that the sheets are nearly as strong as pure graphene. The trick, they write in a paper published in Science, lies in the angles at which the individual sheets are stitched together.
"When you have more defects, you expect the strength to be compromised," said Vivek Shenoy, professor of engineering and the paper's corresponding author, "but here it is just the opposite."
The finding may propel development of larger graphene sheets for use in electronics, optics and other industries.
Graphene is a two-dimensional surface composed of strongly bonded carbon atoms in a nearly error-free order. The basic unit of this lattice pattern consists of six carbon atoms joined together chemically. When a graphene sheet is joined with another graphene sheet, some of those six-carbon hexagons become seven-carbon bonds -- heptagons. The spots where heptagons occur are called "critical bonds."
The critical bonds, located along the grain boundaries, had been considered the weak links in the material. But when Shenoy and Rassin Grantab, a fifth-year graduate student, analyzed how much strength is lost at the grain boundaries, they learned something different.
"It turns out that these grain boundaries can, in some cases, be as strong as pure graphene," Shenoy said.
The engineers then set out to learn why. Using atomistic calculations, they discovered that tilting the angle at which the sheets meet -- the grain boundaries -- influenced the material's overall strength. The optimal orientation producing the strongest sheets, they report, is 28.7 degrees for sheets with an armchair pattern and 21.7 degrees for sheets with a zigzag layout. These are called large-angle grain boundaries.
Large-angle grain boundaries are stronger because the bonds in the heptagons are closer in length to the bonds naturally found in graphene. That means in large-angle grain boundaries, the bonds in the heptagons are less strained, which helps explain why the material is nearly as strong as pure graphene despite the defects, Shenoy said.
"It's the way the defects are arranged," Shenoy said. "The grain boundary can accommodate the heptagons better. They're more relaxed."
Rodney Ruoff from the University of Texas-Austin's Department of Mechanical Engineering is a contributing author on the paper. The National Science Foundation and the Semiconductor Research Corporation's Nanoelectronics Research Initiative funded the research.
- Rassin Grantab, Vivek B. Shenoy, and Rodney S. Ruoff. Anomalous Strength Characteristics of Tilt Grain Boundaries in Graphene. Science, 12 November 2010 330: 946-948 DOI: 10.1126/science.1196893
Cite This Page:
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Banana Theme | Fruit Crafts | Counting | Alphabet Letter B | Preschool Activities Printable Activities
Visit Champ the monkey to count the bananas and also
learn about the benefits of eating bananas!
Activity: Online story and activity with Champ the Monkey
Online story > Bananas for Lunch > Skills > Number order 1-10
This cute story of a monkey eating ten bananas. The story helps children learn and count numbers one to ten.
Online activity > Count the Bananas > Skills > Numbers > Counting and number recognition 1 to 4 | Color > Yellow
Take the children to count the Bananas with Champ the Monkey. Allow the children to roll the mouse over the answers to hear the number. Children may need some help with the activity depending on their age. Make sure to count together and discuss that ripe bananas are yellow.
Activity > Learn about Bananas the Most Popular Fruit in the U.S.
Visit the link above for a presentation to learn about the fascinating history of this delicious fruit.
Activity > Alphabet Letter B Banana printable activities
Activity > Fun in the Kitchen with Bananas > Make a simple recipe with bananas: a smoothie, banana with peanut butter sandwich.
Activity: Fruits Basket Craft > This easy printable craft features an apple, banana and pear.
Letter B Banana printable activities
Fruits Basket Craft
*glue or glue stick
*something to color with
Serve bananas as snack or a simple recipe.
Banana coloring page
|Alphabet > Letter B > Banana | Animals > Monkey | Colors > Yellow | Crafts > Fruits Basket Craft | Holidays & Events > Mar > Nat'l Nutrition Month - Jun > Fruit & Vegetables Month | Online story | activity > Numbers > Counting, number recognition | Nutrition > Fruits|
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Read through the chapters and then go over them again and write out what you notice in your notebook. You can use the following to guide you if you wish. If you don’t like drawing, then just write out the words in appropriate sizes or shapes.
- Draw a map of Egypt and label where Goshen was.
- Write out what Joseph told his brothers to tell Pharaoh their occupation was.
- Draw Jacob, an old man, standing in front of Pharaoh. Write out what Pharaoh asks him and how he responds.
- As the famine continued in the land, how did Joseph have the people pay for the food he gave them in verses 15-21.
- Write out what Joseph gave to the people, and what he claimed for Pharaoh.
- Write out what Jacob asked Joseph to do for him when he died.
Challenge #1: Genesis 46 & 47 PDF
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31 January, 20:27
# A baseball slugger hits a pitch and watches the ball fly into the bleachers for a home run, landing h = 5.5 m higher than it was struck. When visiting with the fan that caught the ball, he learned the ball was moving with final velocity vf = 32.4 m/s at an angle θf = 25.5° below horizontal when caught. Assume the ball encountered no air resistance, and use a Cartesian coordinate system with the origin located at the ball's initial position. a) create an expression for the ball's initial horizontal velocity, V0x, in terms of the variables given in the problem statement. b) calculate the ball's initial vertical velocity, V0y, in m/s. c) calculate the magnitude of the ball's initial velocity, v0, in m/s. d) find the angle, theta0, in degrees above the horizontal at which which the ball left the bat.
+3
1. 31 January, 21:35
0
a) create an expression for the ball's initial horizontal velocity, V0x, in terms of the variables given in the problem statement.
v0x = vf * cos (Θf)
b) calculate the ball's initial vertical velocity, V0y, in m/s
v0x = 32.4m/s * cos (-25.5º) = 29.2 m/s
tanΘ = v1y / v0x → tan (-25.5) = v1y / 29.2m/s → v1y = - 13.93 m/s
the vertical velocity when the ball was caught.
(v0y) ² = (v1y) ² + 2as = (-13.93m/s) ² + 2 * 9.8m/s² * 5.5m = 301.78 m²/s²
v0y = 17.37 m/s
c) calculate the magnitude of the ball's initial velocity, v0, in m/s
v0 = sqrt (v0y^2 + v0x^2)
v0 = sqrt (17.37^2 + 29.2^2) m/s
v0 = 33.98 m/s
d) find the angle, theta0, in degrees above the horizontal at which which the ball left the bat.
tan Θ = v0y/v0x
Θ = arctan (17.37/29.2) = 30.75º above horizontal
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# Homework Help: Differences? Pembezaan
1. Feb 2, 2006
### Mmx
Sorry my maths question is in Malay ill try to translate as gd as i can. I realyl have no idead how to do it.
A zink which have a square size of 30 cm x 16 cm.
At the end of the 4 edge of the sqaure zink is cut out equally same sides x cm.
After cutting the zink is fold into a a open box.
a) Prove that the volume, V of this box is equal to V=4(x^3 - 23x^2 + 120x)cm^3
b) After that, find the maksimum volume, V
If you can understand then nvm cause my maths are in malay and not english. In malay this topic are call Pembezaan im not sure in english that is call differences. I can understand better with the look of working. Sorry
2. Feb 2, 2006
### VietDao29
Okay, I think I understand what you mean...
Do you know the formula for computing the volume of a box: $$\mbox{V = length x height x depth} = \mbox{A_{base}} \mbox{ x height}$$?
So after you fold that zink into an open box, what's the length, the depth, and the height of that box?
So, your zink will look like this:
__|¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|__
|...............................|
|...............................|
|...............................|
|...............................|
¯¯|_______________|¯¯
Hint, the base is the rectangle in red. Can you find its area?
Can you find the height of the box?
From there, can you find its volume?
--------------
For number 2, do you know how to differentiate?
For example f(x) := x3 + x
=> f'(x) = 3x2 + 1.
Can you do this?
If yes, then do you know how one can find an maximum value and minimum value of a function by differentiation?
Last edited: Feb 2, 2006
3. Feb 4, 2006
### Mmx
Thx. Yes i know how to differentiate thats y my malay call Pembezaan. Thx for the hint. I Can do it now.
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## What is the distribution of the interarrival times of a Poisson process?
These “interarrival” times are typically exponentially distributed. If the mean interarrival time is 1/λ (so λ is the mean arrival rate per unit time), then the variance will be 1/λ2 (and the standard deviation will be 1/λ ).
## Is Poisson process continuous-time?
Definition 5.1.3 The Poisson process is one of the simplest examples of continuous-time Markov processes. (A Markov process with discrete state space is usually referred to as a Markov chain).
What is a Poisson process in stochastic process?
A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process that was described in Section 1.3.
### Are interarrival times independent?
By construction, each interarrival time, Xn = tn − tn−1, n ≥ 1, is an independent exponentially distributed r.v. with rate λ; hence we constructed a Poisson process at rate λ.
### How is Poisson process calculated?
Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is: P(x; μ) = (e-μ) (μx) / x! where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.
How do you prove a process is Poisson?
A counting process (N(t))t≥0 is said to be a Poisson process with rate λ, λ > 0, if: (PP1) N(0) = 0. (PP4) The process has stationary and independent increments. (PP5) P(N(h)=1)= λh + o(h).
#### Is Poisson a process?
A Poisson Process is a model for a series of discrete event where the average time between events is known, but the exact timing of events is random . The arrival of an event is independent of the event before (waiting time between events is memoryless).
#### What is a Poisson rate?
The Poisson distribution is defined by the rate parameter, λ, which is the expected number of events in the interval (events/interval * interval length) and the highest probability number of events. We can also use the Poisson Distribution to find the waiting time between events.
How do you calculate interarrival time?
Usually, the timing of arrivals is described by specifying the average rate of arrivals per unit of time (a), or the average interarrival time (1/a). For example, if the average rate of arrivals, a = 10 per hour, then the interarrival time, on average, is 1/a = 1/10 hr = 6 min.
## Is Poisson process ergodic?
Consider the so-called “homogeneous Poisson process”, that is, the classical Poisson process on the real line with intensity equal to the Lebesgue measure. The base transformation is the translation T : x ↦→ x + 1 (in particular, the Poisson T-point process is ergodic).
## Is Poisson process a renewal process?
A Poisson process is a renewal process in which the inter-arrival times are exponentially distributed with parameter λ.
How to find distribution of inter arrival times in Poisson process?
Let X 1 denote the time of first arrival in a Poisson process of rate λ. Let X 2 denote the time elapsed between the first arrival and the second arrival. We can find the distribution of X 1 as follows:
### How is a Poisson process used in continuous time?
A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process that was described in Section 1.3.5. For the Bernoulli process, the arrivals
### Which is a random variable in a Poisson process?
The number of arrivals in an interval of length t is Pois ( λ t) random variable. The number of arrivals that occur in disjoint time intervals are independent of each other. Let X 1 denote the time of first arrival in a Poisson process of rate λ. Let X 2 denote the time elapsed between the first arrival and the second arrival.
How are Poisson processes used in discrete stochastic processes?
A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process that was described in Section 1.3.5.
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- General is defined as something most common, usual, most used or vague.
- An example of general used as an adjective is a general election where voters can vote for anyone they choose.
- An example of general used as an adjective is someone speaking in a general way, a way that most people will understand.
- The definition of a general is a military officer.
An example of a general was former president Dwight Eisenhower who was a five-star general in the U.S. Army.
- of, for, or from the whole or all; not particular or local: a general anesthetic, the general welfare
- of, for, or applying to a whole genus, kind, class, order, or race: the general classifications of matter
- existing or occurring extensively; common; widespread: a general unrest
- most common; usual: the general spelling of a word
- concerned with the main or overall features; lacking in details; not specific: the general features of a plan
- not precise; vague: to speak in general terms
- senior or highest in rank: a general manager, an attorney general
- not connected with or limited to one branch or department of learning, business, etc.; not specialized: a general store
Origin of generalMiddle English ; from Old French ; from Classical Latin generalis ; from genus (gen. generis), kind, class: see genus
- the main or overall fact, condition, idea, etc.
- the head of some religious orders
- Archaic the public; populace
- any of various military officers ranking above a colonel; specif.,
- such an officer, with an insignia of four stars, ranking above a lieutenant general
- U.S. Marine Corps an officer of the highest rank
- an anesthetic that makes a patient unconsciousin full general anesthetic
- in the main; usually
- without specific details
- with reference to all spoken of
- Concerned with, applicable to, or affecting the whole or every member of a class or category: “subduing all her impressions as a woman, to something more general” (Virginia Woolf).
- Affecting or characteristic of the majority of those involved; prevalent: general discontent.
- Of or affecting the entire body: general paralysis.
- Being usually the case; true or applicable in most instances but not all: the general correctness of her decisions.
- a. Not limited in scope, area, or application: as a general rule.b. Not limited to or dealing with one class of things; diversified: general studies.
- Involving only the main features rather than precise details: a general grasp of the subject.
- Highest or superior in rank: the general manager.
- a. A commissioned rank in the US Army, Air Force, or Marine Corps that is above lieutenant general.b. One who holds this rank or a similar rank in another military organization.
- A general officer.
- A statement, principle, or fact that embraces or is applicable to the whole.
- General anesthesia.
- Archaic The public.
Origin of generalMiddle English, from Latin generalis, from genus, gener-, kind; see gen&schwa;- in Indo-European roots.
(comparative more general, superlative most general)
- Including or involving every part or member of a given or implied entity, whole etc.; as opposed to specific or particular. [from 13th c.]
- Applied to a person (as a postmodifier or a normal preceding adjective) to indicate supreme rank, in civil or military titles, and later in other terms; pre-eminent. [from 14th c.]
- Prevalent or widespread among a given class or area; common, usual. [from 14th c.]
- Not limited in use or application; applicable to the whole or every member of a class or category. [from 14th c.]
- Giving or consisting of only the most important aspects of something, ignoring minor details; indefinite. [from 16th c.]
- Not limited to a specific class; miscellaneous, concerned with all branches of a given subject or area. [from 16th c.]
- Commander of an army.
- Hannibal was one of the greatest generals of the ancient world.
- (military) A rank in the army and air force that is higher than colonel or brigadier, and is usually the highest rank group next to commander in chief, except in countries that use the rank of field marshal.
- (military) a commissioned rank in the British Army and Royal Marines, above lieutenant general and below field marshal.
- (military) a commissioned general officer in the United States Army, Marine Corps, or Air Force superior to a lieutenant general. A general is equal in rank or grade to a four star admiral. In the US Army, a general is junior to a general of the army. In the US Marine Corps, a general is the highest rank of commissioned officer. In the US Air Force, a general is junior to a general of the air force.
- Short for general anaesthetic or general anaesthesia.
When used as a title, it is always capitalized.
- Example: General John Doe.
The rank corresponds to pay grade O-10. Abbreviations: GEN.
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All in the Mind
Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?
Rotating Triangle
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Instant Insanity
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
Threesomes
Here is a synopsis of the solutions offered for the cases considered so far (i.e. it does not consider triangles that have non-horizontal bases):
The smallest triangle it is possibkle to draw has a base of 1 unit and a height of 1 unit. So the smallest area is $\frac{1}{2}$ sq. unit.
There are an infinite number of triangles that can be drawn with these diagonals (see the problem "Shear Magic" )
There are two ways of creating a triangle of area 1 sq and with a horizontal base:
Base 1 unit; height 2 units
or
Base 2 units and height 1 unit, again
For an area of 2 sq units there are three families of triangles with a hoirizontal base::
Base 1 unit and height 4 units
or
Base 2 units and height 2 units
or
Base 4 units and height 1 unit
For each family there are an infinite number of triangles
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Chickenpox Vaccine for Grown-Ups, TooLOADING...
If you’re an adult, chances are you had chickenpox when you were a kid. But if you didn’t, maybe you should think about getting the chicken pox vaccine.
The chickenpox vaccine became available in the United States in 1995 and since then has been mostly regarded as a vaccine for children. The Centers for Disease Control and Prevention’s (CDC) Advisory Committee on Immunization Practices, the American Academy of Pediatrics and the American Academy of Family Physicians recommend all children be routinely vaccinated between 12 and 18 months of age and that all susceptible children – those with weak immune systems – receive the vaccine before their 13th birthday.
The CDC also recommends the chickenpox (also called varicella) vaccine for susceptible teenagers and adults, especially health care workers. But information presented at an annual meeting of the Infectious Diseases Society of America suggests even healthy adults who managed to escape chickenpox as a child could benefit from the vaccine. That’s because adults are more likely to have a more serious case of chickenpox with a higher rate of complications.
Getting chickenpox can be serious
“Older people are more susceptible to complications of chickenpox, such as pneumonia and central nervous system infections,” says Krow Ampofo, M.D., fellow in pediatric infectious diseases, Columbia Presbyterian Hospital in New York City, who presented results of a study at the meeting. “Yet, individuals in this age group are less likely to be vaccinated than children.”
Chickenpox, which is caused by the varicella zoster virus, is mostly known for its characteristic itchy rash, which then forms blisters that dry and become scabs in four to five days. The virus is spread from person to person by direct contact or through the air. About 90 percent of people in a household who have not had chickenpox will get it if exposed to an infected family member.
Symptoms of chickenpox may begin at any time within seven to 21 days after exposure, with most cases appearing between 14 and 17 days.
How effective is the vaccine?
The chickenpox vaccine has been shown to be less effective than other children’s vaccinations. In fact, about 10 percent to 30 percent of children who receive the vaccination do not become immune to chickenpox.
Ampofo studied 557 vaccinated adults ages 22 to 46. Of that group, 43 (8 percent) developed chickenpox.
The chickenpox vaccine has been proven to be highly effective in protecting against severe chickenpox, and Ampofo says that also was borne out in his study.
“Among those who did contract chickenpox, the disease was generally mild,” he says.
Some people who get chickenpox suffer from complications, such as fluids (dehydration), pneumonia, meningitis, inflammation of the heart or Reye’s syndrome.
None of the adults in Ampofo’s study had significant adverse effects from the vaccination itself. “A few developed mild rashes within six weeks that went away without complications,” Ampofo says.
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Stage 2 | Subject outline | version control
Accredited in May 2015 for teaching at Stage 2 from 2017.
Stage 2 | Subject outline | Content | Creating texts
Students create a range of texts for a variety of purposes. By experimenting with innovative and imaginative language features, stylistic features, and text conventions, students develop their personal voice and perspectives. They demonstrate their ability to synthesise ideas and opinions, and develop complex arguments.
Accurate spelling, punctuation, syntax, and use of conventions should be evident across the range of created texts. Students benefit from modelling their own texts on examples of good practice in the same text type. In creating texts students extend their skills in self-editing and drafting.
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# Lec 7 Implementasi Disjoint Set
Gratis
0
0
29
6 months ago
Preview
Full text
Making a “good” maze
Up-trees
Maze revisited
Represent maze environment as graph {V,E} collection of rooms: V
connections between rooms (initially all closed):
### E
Construct a maze: collection of rooms: V = V
designated rooms in, iV, and out, oV
collection of connections to knock down: E E such that a unique path connects any two rooms
So far, some walls have been knocked down while others remain.
Now, we consider the wall between A and B.
A
Should we knock it down? no, if A and B are otherwise connected
B yes, if A and B are not otherwise connected While edges remain in E
Remove a random edge e = (u, v) from E
If u and v have not yet been connected
e to E
u and v as connected
• mark
An equivalence relation R has three properties: refexive: for any x, xRx is true symmetric: for any x and y, xRy implies yRx transitive: for any x, y, and z, xRy and yRz implies xRz
Connection between rooms is an equivalence relation (call it C)
For any rooms a, b, c a C a (A room connects to itself!) If a C b, then b C a If a C b and b C c, then a C c
Other equivalence relations? Union/Find ADT operations union find( 4 )
{1,4,8} {6} fnd
create
8
{7}
{2,3,6} destroy
{5,9,10}
{2,3} union(
2 ,
6 )
Disjoint set equivalence property: every element of a DS U/F structure belongs to exactly one set Dynamic equivalence property: the set of an element can change after execution of a union
Given a set U = {a 1 , a 2 , … , a n } Maintain a partition of U, a set of subsets of U {S 1 , S 2 , … , S k } such that: -
• - each pair of subsets S i and S j are disjoint: together, the subsets cover U:
• - The S i
are mutually exclusive and collectively exhaustive Union(a, b) creates a new subset which is the union of a’s subset and b’s subset
NOTE: outside agent decides when/what to union - ADT is just the
bookkeeper
Find(a) returns a unique name for a’s subset NOTE: set names are arbitrary! We only care that:
k i i S U 1
j i S S
3
10
a b c
Construct the maze on the right
2
1
6 Initial state (set names underlined):
4
7
d e f
{a}{b}{c}{d}{e}{f}{g}{h}{i}
11
9
8 Maze constructor (outside agent) traverses edges in random order
12
5 and decides whether to union
g h i
{a}{b}{c}{d}{e}{f}{g}{h}
3
10
a b c
{i}
2
1
6 find(b) b find(e) e
4
7
d e f
1 to E
11
9
8 union(b, e)
12
5
g h i
{a}{b,e}{c}{d}{f}{g}{h} {i}
Order of edges in blue
Finding the representative member of a set
is somewhat like the opposite of finding whether a given item exists in a set.
So, instead of using trees with pointers from each node to its children; let’s use trees with a pointer from each node to its parent.
Each subset is an up- tree with its root as its representative member
a c h
g
All members of a given set are nodes in that set’s up-tree
f i d b
Hash table maps input data to the node associated with that
e
data Up-trees are not necessarily binary! find( f ) find( e )
a b c
10
a c g h
d e f
7
f i
d b
11
9
8 g h i
12 e
Just traverse to the root! union( a,c )
a b c
10 a c g h d e f
f i d b
11
9
8 g h i
12
e
Just hang one root from the other! runtime: a b c
3
10
2
1
6 d e f
4
7
11
9
8
union( b , e )
g h i
12
5
a b c d e f g h i a b c d f g h i a b c
3
10
2
6 d e f
4
7
11
9
8
union( a , d )
g h i
12
5 a b c d f g h i
e a b c f g h i a b c
3
10
6 d e f
4
7
11
9
8
union( a , b )
g h i
12
5 a b c f g h i
d e a c f g h i b d a b c
10
6 d e f
4
7
11
9
8
find( d ) = find( e )
g h i
12
5 No union!
a c f g h i b d
e a b c
10
6 d e f
7
11
9
8
union( h , i )
g h i
12
5
a c f g h i a c f g h b d i b d a b c
10
6 d e f
7
11
9
8
union( c , f )
g h i
12
a c f g h a c g h b d b d f i i a b c
10 d e f
7
find( e )
11
9
8
find( f ) union( a , c )
g h i
12 a c g h
c g h b d f a f i i b d e a d b e c f g h i
11
10
9
8
12
f g h a b c i d f g h a b
c
i d find( f ) find( i ) union( c , h ) find( e ) = find( h ) and find( b ) = find( c ) So, no unions for either of these.
a d b e c f g h i
11
10
9
12
f g h a b
c
i d a b c d e f
find( d )
11
find( g )
g h i
union( c , g )
12 c g
g c a f h a f h b d i b d i find( g ) = find( h ) So, no union.
And, we’re done!
a d b e c f g h i
12
f g h a b c i d
a d b e c f g h i
Ooh… scary!
A forest of up-trees can easily be
a c g h stored in an array.
Also, if the node
b d f i
names are integers or characters, we can use a very
e
simple, perfect Nifty storage trick! hash.
0 (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 (i)
up-index: -1 -1
1 2 -1 -1
7
typedef ID int;
ID union(ID x, ID y) {
ID find(Object x) { assert(up[x] == -1); assert(hTable.contains(x)); assert(up[y] == -1);
ID xID = hTable[x]; up[y] = x; while(up[xID] != -1) { } xID = up[xID]; } return xID; } runtime: O(depth) or … runtime: O(1)
Simple ADT, simple data structure, simple code
Complex complexity analysis, but extremely useful
result: essentially, constant time!
Lots of potential applications Object-property collections Index partitions: e.g. parts inspection To say nothing of maze construction
In some applications, it may make sense to have meaningful (non-arbitrary) set names
Gratis
## Tags
7 Set 14 46 7 Set Implementasi Zona Set Implementasi Data Mining Rough Set Dalam Pengumuman Pl Set 4 6 Pengumuman Pl Set 4 6 7 7 Penjelasan Dokumen Pemilihan Implementasi Lec 06 Frame Based Expert Systems Lec 02 Rule Based Expert Systems Power Set Of Fuzzy Set Pengumuman Set 41 Pengumuman Set
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Table of Contents
Identity is a widely used term that means different things to various groups of people at different times. Identity is sometimes taken to mean a sense of integration of self, in which different aspects are unified (Deaux and Martin 103). Identity can also be thought to be a way in which individuals or groups see and define themselves, and how other individuals or groups see and define them. Identity is formed through the socialization process and the influence of social institutions like the family, the education system, and the mass media (Browne, 5). Music represents one of the factors that greatly influence the identity of person or group. From its sound and setting to dance and instrumentation, music represents a powerful symbol and a tool for creating and maintaining identity. This essay explores the Igbo folk music with a view of establishing how it gives identity to the Igbo community.
Music and identity
Music is a fundamental channel of communication as it provides a means through which people can share emotions, intentions, and meanings, even when their spoken languages differ. It also provides a vital lifeline to human interaction, especially those who have special needs or interests. According to Connell and Gibson (15), music is an extremely powerful tool through which people develop personal and social identities. It represents a channel through which emotions, thoughts, political statements, social relationships, and physical expressions are exchanged.
Sieber (123) points out that popular music is a powerful medium for representing, contesting, and negotiating the changing cultural identities in the dynamic global diasporas. It indexes continuity and change in society. It also sustains and renegotiates connection across transnational space. Lastly, music reshapes generational relations by linking the present and past generations. In a nutshell, music is used as a communication means through which aspect of people’s identity as constructed. Regardless of its genre, music has a profound influence on developing a sense of peoples’ identity, values, and beliefs.
The Igbo folk music and identity
Folk music is a spontaneously composed type music for a race, tribe, or group (Nnamani 304). It is humble in nature and is orally transmitted from generation to generation. It has unknown composers. According to Nnamani (304), Folk music is the term used to designate the traditional music of a person or group, which contrasts the popular music and the serious concert or opera music.
The Nigerian music is conceived as a medium of aesthetic contemplation of socio-cultural phenomenon (Omojola par. 1). Its importance transcends the value created by itself. This is because the composer of the music, its performers, and audience attach premium value to its relevance to the socio-political issues that affect their daily lives. The history and development of the Igbo music date back to the traditional African society before the scramble for Africa by the European states. The Igbo music was formulated on communal binding and viable traditional concepts and covenants (Omojola par. 6). These concepts were periodically validated or commemorated as a means of binding generations together. The generational binding required to be facilitated by stylized media that could give a super ordinary atmosphere when passed from one generation to another (Omojola par. 7). The media for facilitation of these functions consisted of traditional African theater that was wide in scope and ramification. Music was the common media used to pass cultural norms, beliefs, and values from one generation to another. It included dances, drama, and mime. These music aspects were perfected with time to become part of the Igbo culture.
Music is an important aspect of the life of the Igbo people because it is known to possess both cultural and spiritual values. Celestina and Veronica (269) posit that music accompanies the life of a black man from the womb to the tomb. They assert that music in the Igbo community dictates the life of a person. It represents a celebration the birth of a child; it is used during children’s games and at peer group functions. Also, it is used during work and leisure time, religious functions, and at death. Children are exposed to music from infancy through lullabies (Celestina and Veronica 269). Also, an Igbo child is exposed to folk songs and games performed by fellow children during playtime. At these stages of life, the performed music affects the identity of the child into adulthood. To the Igbo people, folk songs form an important channel through which proverbs and idioms are passed. It is through the folk music that virtues are upheld and vices condemned. The folktales and music stress the importance of observing the social norms.
Music has been as a source of identity for the Igbo people in diaspora. The performance of Igbo music in the US has helped reinforce the culture and identity of the Igbo community. The performers and audience of the Igbo music are of different age groups. The singing and performances of the songs are done by people from different ages to create social, emotional, and aesthetic relationships and solidarity (Gordon 238; Nuraghe and Frank 229). Children sing the Igbo songs as a sense of attachment to the culture and traditions of the community. The Igbos sing and dance to propel their philosophy of life, wisdom, and sentiments. The Igbo music and dance use different instruments. These include the “Igbo” or simply the big drum, the “One”, flutes, and the “Ekwe” (Ndukaihe and Fonk 229; Omojola par. 7). The move makes the Igbo music gain an Afrobeat style of music, which is a source of Nigerian identity.
It is worth to acknowledge that the Igbo folk music is used to serve a social and cultural course. It is used to preserve and propagate the Igbo cultural heritage, which is a source of their identity (Ndukaihe and Fonk 229). It is used to express the Igbo language, which they consider as a lovable language. The melodic style takes permanent residence in the cultural practices of the community because of its intrinsic quality (Gordon 238). The melodies and the rhythmic waves of the Igbo music forge mystical and aesthetic links among the performer and audience. It creates an identity for the tribe, making the Igbo people have a distinct identity formation as compared to the other tribes (Celestina and Veronica 269).
Although modernity and distance have caused generations to lose touch of the Igbo culture, the modern performances of the Igbo folk music at the national and international levels has enabled the Reformation and observation of the Igbo identity. For instance, folk music has given the Igbo people living in the US a sense of identity for a long time. According to Gordon (238), folk music helps the immigrant persons living in large cities across the US to reinforce their cultural, tribal, and religious identities. Igbo is one of the prominent tribes in Los Angeles that has propelled its identity through folk music. Gordon (238) observes that the Igbo people do not feel at home with the mode of worshipping in the American churches. Instead, the Igbo sing and perform Igbo songs in churches. Women wear the traditional Igbo attire during the performances, which are usually accompanied by drumming. Gordon (238) notes that this has helped to lessen the sense of isolation among the Igbo community. It has also given them the identity that has enabled them to deal with the pressure of living away from home.
The Expression of Chineseness and Americanness
Research makes it clear that as from the 1990s, Chinese mainstream music witnessed an expansion in the quantity of Chinese American or American-Born Chinese (ABC) artists that left the United States to seek after professions in China. These artists who did not have their cultural legacy in China, additionally carry with them parts of the American experience. Two of the most unmistakable ABC pop stars in China are Wang Leehom and Vanness Wu. However, despite the fact that the two have comparable foundations and both recognize as ABC, the routes in which they express their Chineseness and Americanness in their music is open (Boxi, Pg. 73). One would thus be right to proclaim that one may make use of music to promote a secure identity in the community.
It merits noting that the typical Asian American experience comprises of immigrants from Asia in pursuit of more and possibly better opportunities in the community. It deserves noting that in most cases, they usually gain exposure to Asian American music which may comprise of both Asian America and music that it created by the Asians in America. It deserves noting the interactions of the different cultural identities would also give way to the introduction of the Asian American music in Asia.
It is evident that as from the 1990s, the traditional Chinese music would receive influence from the increasing number of number of Chinese who were undertaking musical activities in America. The move would offer them a peculiar identity in the Chinese world. Some musicians rose to prominence due to the integration of the American way of life, and the Chinese culture includes Wang Leehom and Vaness Wu (Boxi, Pg. 73). It merits noting that music would offer them an individual identity in the community due to the nature of their music.
with any paper
It is apparent that as from the 1980s, Mandopop was introduced to Taiwan and would experience and exponential growth. The Taiwanese artists highly dominate the music genre. It deserves noting the origin of the style of music does not have a significant influence on its popularity. Research makes it clear that music has more financial goals to attain than the political mileages. The move makes it possible for the genre of music to appeal to a relatively broad audience in the community. The style attracts followers from as far as South East Asia and Eastern Asia. An insight into the music genre makes it vivid that the western culture had a significant impact on the development of the music genre. The influence is likely to rise due to globalization and transnationalism (Boxi, Pg. 73). It is possible to trace the initial stages of the influence from 1960 when the American dance band’s style of instrumentation would become prominent in Taiwan.
The influence of the western cultures would intensify with the return of the American born Chinese. Transnationalism would make it feasible for them to bring in their American experience. It merits noting that the desire to return to one’s homeland encourages the Chinese immigrants in the United States to maintain closer ties with the native homeland. The move makes a majority of the American Chinese to consider returning to their home. It deserves noting that the move has an impact on their cultural heritage since the majority of them already have experience of the American way of life.
There is economic point of views that make an effort to exploit the possible causes of the rise of the Mandopop. It merits noting that there have been increases in the cases Asians going back to their homelands in pursuit of better opportunities. It was the case for some aspiring Chinese musicians who were in America. They would bring in the American musical qualities such as the hip-hop rhythm to the famous Chinese music. It is such a scenario that would give rise to the LA Boyz hip hop group in China. It merits noting that the team would bring in a different vibe to the Chinese music. They would introduce the wearing of the baggy jeans, rapping and the ability to carry out street dances. The move would highly popularize the American hip hop in China (Boxi, Pg. 74). The success of the music group was a great inspiration to the American born Chinese to seek an opportunity out of America.
An insight to the rise of Mandopop would be incomplete without an overview of Wang’s and Vanessa’s contributions. Their music was essential in offering them an identity in the community. They portray some difference which is crucial in their bid to create their musical identities. The difference stems from the level of Chineseness and Americanness that they opt to corporate in their music. It merits noting that Wang would opt to create a distinct Chinese pop sound which he would describe as chinked out. He would infuse some Chinese elements in his music such as the Chinese musical instruments. He was also keen to maintain his pop music base which would in some occasions contain rap interludes. It deserves noting that there was a considerable variation of English in the songs (Boxi, Pg. 74). It is also evident that in instances when Wang uses the English phrases, in most cases, they tend to be colloquial.
An analysis of the visual representation of the level of Chineseness and Americanness reveals a considerable variation in their representation. The move offers Wang’s music a peculiar identity in the community. Wang appears with two women in one of the videos where one of the women is in modern clothing while the other one adorns traditional Chinese clothing. The move may seem appropriate in the attempt to offer an equal representation of both cultures.
The public persona of Wang reveals a blend of the two cultures. His dressing mode which consists of jeans and jackets could have its origin in America. However, then move to make use of his Chinese name makes it clear that he embraces the Chineseness in him. The move makes it possible for him to relate well with his audience.
Margins and Mainstreams in Jazz music
It deserves noting that throughout history, there has been a tendency to view jazz music as marginal in the community. There are concerns that the genre of music enjoys neither the straightforward commercial nature of rock music in the community not the level of public of public support accorded to the classical music. It is apparent that the term jazz loosely translates into sexual intercourse. The name was a never a great start for the introduction of the genre in the community (Stranbridge, Pg. 2). It is apparent that music deserves to be ideal for the mainstream audience in the community. There are few claims that jazz would rise to be a mainstream music in the community.
A deeper insight into the struggling nature of the jazz music in the community reveals that its history is a significant threat to its penetration in the community. The swing era would mark the final moments of the fading popularity of the jazz music. The music was becoming esoteric to the mainstream tastes. It was increasing becoming inappropriate to the mass. The socio-cultural interactions were behind the lagging nature in the acceptance of the music genre in the community. There was a tendency to link the music genre to prostitution and alcoholism in the community.
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It merits noting that jazz music would occur a different position in the North American cultures after the Second World War. There was a significant consideration of its outside role with the increasing exposure to the mainstream audience in the community (Stranbridge, Pg. 2). The period between 1950 and 1960 would witness various recording in the genre. One of the important albums from that period is the album Time Out which happens to be among the most sold jazz albums in the community. It is also the same period where there was a relatively wider acceptance of the jazz music in the community. Several Hollywood movies would feature jazz tracks which would lead to greater acceptance of the music genre in the community. Some television series would also feature jazz tracks. The move was commendable as it would aid in the penetration of the jazz music in the community. One would be right to proclaim that the period between 1950 and 1960 was the jazz age (Stranbridge, Pg. 2). The society would accept the genre despite its initial negative publicity.
The period after the Second World War is critical in the analysis of the musical identity of jazz music in the community. There was a popular populist conception of the musical genre which had links to the jazz canon. The conceptualization would also serve as a great marketing strategy for the music in the community. It deserves noting that the populist notion has links with connotations of sophistication which were essential in eroding the initial stereotypes that the community held against the music genre. The move was pivotal in the attempt to influence the masses to accept the music genre widely.
There is essence in noting that apart from the sharp contrast in the discursive role of jazz music in the community, there are more challenging issues relating to the genre of music. Some of the problems would exhibit different relationships with the existing cultural mainstream due to the oppositional politics which would emerge from the notions of marginality of the music genre. There are circles where the jazz music holds the locus of discussion on the possibility of the roles that it may play in promoting social change in the community (Stranbridge, Pg. 2). There is a potential link between the jazz music and social, political benefits in the community. It merits noting that in cultural policy and arts funding in the community, there are several economic benefits such urban regeneration, promotion of tourism in the economy, business investments and also the development of service industry which is the backbone of the economy in certain states. It also merits noting that embracing the music genre could have a significant impact on the unemployment in the economy. It is likely to lead to job creation thus minimizing unemployment.
It merits noting that arts have a profound impact on the social activities in the community. Jazz is one of the forms of art in the community. It offers the society a full range of benefits from economic to entertainment. However, there is a certain degree of distraction that one may link the genre jazz within the community. The pragmatic blend of art and culture is essential in the attempt to have the genre receive acceptance in the community.
Music is an important anthropological subject that has gained prominence in different fields. Its importance surpasses the entertainment, singing, and dancing constructs. Music offers an intrinsic value based on the fact that it influences the identity of a person, group, or national. The case of the Igbo folk music illustrates that music determines the cultural heritage and identity. The folk music has given the Igbo tribe the identity, making it stand out among the Nigerian tribes. It has also upheld the cultural and religious identities of the Igbo people in the diaspora, especially in the US.
- Browne, Ken. Introducing sociology for AS level. Malden, MA: Polity, 2006. Print.
- Celestina, Esimone C, and Ojukwu E. Veronica. “Music as an Instrument of Identity and Cultural Heritage Preservation: A Study of the Igbo Tribe in Nigeria”. Journal of Teaching and Education 3.3 (2014): 269-275. Print.
- Connell, John, and Chris Gibson. Sound Tracks: Popular Music, Identity, and Place. New York, NY: Psychology Press.
- Deaux, Kay, and Daniela Martin. “Interpersonal networks and social categories: Specifying levels of context in identity processes.” Social Psychology Quarterly 66.2 (2003): 101-117.
- Gordon, April A. Nigeria’s Diverse Peoples: A Reference Sourcebook. Santa Barbara, CA: ABC-CLIO, 2003. Print.
- Ndukaihe, Vernantius E, and Peter Fonk. Achievement as Value in the Igbo/African Identity: The Ethics. Berlin: Lit, 2006. Print.
- Nnamani, Nnamani S. “The Role of Folk Music in Traditional African Society: The Igbo Experience.” Journal of Modern Education Review 4.4 (2014): 304-310. Print.
- Omojola, Olabode. Music Education in Nigeria: Historical Trends. Web. 17 Mar. 2015. <http://www.unilorin.edu.ng/journals/education/ije/dec1992/MUSIC%20EDUCATION%20IN%20NIGERIA.pdf>.
- Sieber, Timothy. “Popular music and cultural identity in the Cape Verdean post-colonial diaspora.” Etnográfica 9.1 (2005): 123-148. Print.
- Stanbridge, Alan. “From the margins to the mainstream: Jazz, social relations, and discourses of value.” Critical Studies in Improvisation/Études critiques en improvisation 4.1 (2008). Print.
- Chen, Boxi. “The Expression of Chineseness and Americanness in Chinese Popular Music: A Comparison of ABC Pop Stars Wang Leehom and Vanness Wu.” Asian Music 43.2 (2012): 71-87. Print
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Electrocardiogram (ECG or EKG) and ECG Stress Test
The electrocardiogram is commonly used as a relatively simple way of diagnosing heart conditions.
The electrocardiogram (ECG or EKG) evaluates the heart's rhythm and electrical activity. An ECG stress test is an ECG that is recorded during exercise-typically on a treadmill.
The ECG (or EKG) is a noninvasive test, ordered by your physician that records the electrical impulses that travel through the heart. This electrical activity determines the heart's rate and rhythm.
The electrical waves recorded by the ECG can help your physician determine whether your heart is functioning normally or experiencing problems. The waves are registered by electrodes placed on the chest, arms and legs. Each electrode controls an ink needle that writes on a grid paper. The higher the intensity of the electric wave, the higher up the needle will move on the paper. The paper moves at a certain speed beneath the needle, resulting in an ink curve.
There is no special preparation involved. You should not apply lotions or oils to your chest prior to the procedure. There are typically no restrictions on food, liquid or medications prior to the test. The ECG takes about 15 minutes.
Prior to Procedure
- You will:
- Have a physical exam and be asked about your medical history
- Have your chest shaved if you have a hairy chest
- For a stress test, you should:
- Allow two hours between your last meal and the stress test
- Wear comfortable clothing and walking shoes
Description of the Procedure
When your heart beats, it creates electrical signals. The ECG detects these signals from the surface of your skin and records them on a piece of graph paper. You will not feel anything during the procedure.
You’ll be asked to lie quietly on your back, with your shirt off. Six to twelve small adhesive pads (no suction cups) with attached wires will be placed across your chest, arms and legs. The wires will connect to the ECG machine. Your heart rhythms will be recorded in a few minutes and the test is complete.
If you’re having an ECG Stress Test, your heart rhythm will be recorded while you exercise, usually on a treadmill. The speed and slope of the treadmill will be slowly increased as you walk. The test will continue until you have reached a certain heart rate, certain ECG changes occur, or you are too tired to continue, are short of breath, or have chest pain. This test typically lasts less then 30 minutes.
Depending on your condition and your doctor’s assessment, you may have to have more tests. If you have a heart condition or abnormal ECG, keep a recent copy of your ECG in your wallet, purse, or car.
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Percentages Practice Set 8
5 Steps - 3 Clicks
Percentages Practice Set 8
Introduction
Percentages: A Percentage is a dimensionless ratio or number expressed as a fraction of 100. It is often denoted by the character (%).
Example: 10% = $$\frac{10}{100}$$ = $$\frac{1}{10}$$
Percentages is one of the important topic in the Quantitative Aptitude section. The article Percentages Practice Set 8 consists of different models of questions with answers useful for candidates preparing for different competitive examinations like RRB ALP/Technical Exams/Junior Engineer Recruitment Exams, SSC, IBPS PO Exams and other examinations across the globe that include Quantitative Aptitude section. Prepare better for all exams with this Percentages Practice Set 8 for SSC CGL & Railways. This Percentages Practice Set 8 for SSC, Railways Exams will help you learn concepts of mensuration.
Quiz
1. A sells his goods 50% cheaper than B but 50% dearer than C. The cheapest is?
A. A
B. B
C. C
D. All Alike
Explanation:
Let B = 100
A = 50
C * ($$\frac{150}{100}$$ ) = 50
3C = 100
C = 33.3 then ‘C’ Cheapest
2. The salary of a typist was at first raised by 10% and then the same was reduced by 5%. If he presently draws Rs.1045.What was his original salary?
A. 900
B. 950
C. 1000
D. 1000
Explanation:
X * $$\frac{110}{100}$$ * $$\frac{95}{100}$$ = 1045
X * ($$\frac{11}{10}$$) * ($$\frac{1}{100}$$) = 11
X = 1000
3. The tax on a commodity is diminished by 20% and its consumption increased by 15%. The effect on revenue is?
A. It increases by 8%
B. It decreases by 8%
C. No change in revenue
D. No change in revenue
Explanation:
100 * 100 = 10000
80 * 115 = 9200
———–
10000———–800
100———–? => 8% decrease
4. A candidate got 35% of the votes polled and he lost to his rival by 2250 votes. How many votes were cast?
A. 7500
B. 5000
C. 6000
D. 3500
Explanation:
35%———–L
65%———–W
——————
30%———-2250
100%———? => 7500
5. Subtracting 10% from X is the same as multiplying X by what number?
A. 80%
B. 90%
C. 10%
D. 50%
Explanation:
X – $$\frac{10}{100}$$ X = X * ?
? = 90%
1. An engineering student has to secure 36% marks to pass. He gets 130 marks and fails by 14 marks. The maximum No. of marks obtained by him is?
A. 300
B. 400
C. 350
D. 500
Explanation:
130
14
——-
361—— 144
100%——? => 400
2. A and B’s salaries together amount to Rs. 2,000. A spends 95% of his salary and B spends 85% of his. If now their savings are the same, what is A’s salary?
A. Rs.500
B. Rs.750
C. Rs.1250
D. Rs.1500
Explanation:
(5/100) A = ($$\frac{15}{100}$$) B
A = 3B
A + B = 2000
4B = 2000 => B = 500
A = 1500
3. 5% people of a village in Sri Lanka died by bombardment, 15% of the remainder left the village on account of fear. If now the population is reduced to 3553, how much was it in the beginning?
A. 3800
B. 4200
C. 4400
D. 5500
Explanation:
X * (latex]\frac{95}{100}[/latex]) * (latex]\frac{85}{100}[/latex]) = 3553
X = 4400
4. The tank full of petrol in Arun’s motorcycle lasts for 10 days, if he starts using 25% more every day for how many days will the tank full of petrol last?
A. 5 days
B. 6 days
C. 7 days
D. 8 days
Explanation:
alcohol = $$\frac{30*2}{5}$$ = 12 and water = 18 litres
Let us assume that Arun uses X units of petrol everyday.
So the amount of petrol in the tank when it is fuel will be 10 X.
If he started using 25% more petrol every day, then the amount of petrol he how uses every day will be
X (1 +$$\frac{25}{100}$$) = 1.25 x
Therefore, number of days his petrol will how last = Amount of petrol in tank / amount of petrol used everyday = $$\frac{10x}{1.25x}$$ = $$\frac{10}{1.25}$$10/1.25 = 8 Days
5. In an examination, every candidate took physics or mathematics or both 65.8% took physics and 59.2% took mathematics the total number of candidates was 2000. How many candidates took both physics and mathematics?
A. 750
B. 500
C. 250
D. 125
Explanation:
Let x% candidates take both the subjects.
Therefore, Percentage of candidates who opted physics = 65.8%
And the percentage of candidates who opted mathematics = 59.2%
Therefore, x =(65.8 + 59.2 – 100)%
= (125 -100)% = 25%
Also the total number of candidates = 2000
Therefore, Number of candidates who opted both the subjects = $$\frac{25}{100}$$ x 2000 =500
1. In a survey it was found that 80% of those surveyed owned a car while 60% of those surveyed owned a mobile phone, if 55% owned both a car and a Mobile phone, What percent of those surveyed owned a car or a mobile phone or both?
A. 65%
B. 80%
C. 85%
D. 97.5%
Explanation:
Given that percentage of car owners = 80%
Percentage of mobile phone owners = 60%
Percentage of people having both car and mobile phone = 55%
Percentage of people having only car = 80 -55 = 25%
Percentage of people having only mobile phone = 60 -55 =5%
Percentage of people having car or mobile phone or both = 55% + 25% + 5% = 85%
2. In a test a candidate attempted only 8 Questions and secured 50% marks in each of the questions if the obtained a total of 40% in the test and all questions in the test carried equal marks, how many questions were there in the test?
A. 8
B. 10
C. 15
D. 16
Explanation:
Let the marks of each question be 10
Total marks got by the candidate = 8 x 5 = 40 marks
40% = 40 : 100 = 100
Therefore, Total number of questions = 10 $$\frac{100}{10}$$ = 10
3. A city has a population of 300000 out of which 180000 are males 50% of the population is illiterate if 70 % of the males are literate, then the number of literate females is
A. 24000
B. 30000
C. 54000
D. 60000
Explanation:
Total population = 300000
Total number of males = 180000
Total literates = 5% of total population = 150000
Number of literate males = 70% of males = 126000
4. In a company, 60% of the employees are men, of this 40 % are drawing more than Rs. 50000 per year. 36% of the total employees of the company draw more than Rs.50000 per year then what is the percentage of women who are drawing less than Rs. 5000 per year?
A. 70%
B. 60%
C. 40%
D. 30%
Explanation:
Total number of employees be 100
Then number of men = $$\frac{6000}{100}$$ = 60
Number of women = $$\frac{4000}{100}$$ = 40
Therefore, a number of men drawing more than Rs. 50000 = $$\frac{24000}{100}$$ = 24 men
Since the number of total employees drawing more than Rs. 50000 = $$\frac{3600}{100}$$ = 36
Number of women who draw more than Rs. 50000 = 36- 24 = 12
Number of women who draw less than Rs. 50000 = 40 -12 = 28
Therefore, the Percentage of women who draw less than Rs. 50000 per year = $$\frac{28}{40}$$ x 100% = 70%
5. In an election between two candidates. One got 55 % of the total valid votes. 20 % of the votes were invalid. If the total number of votes was 7500. The number of valid votes that the other candidate got was
A. 2700
B. 2900
C. 3000
D. 3100
Explanation:
Valid votes = ($$\frac{80}{100}$$ x 7500) = 6000
Valid votes polled by one candidate
= ($$\frac{55}{100}$$ × 6000) = 3300
Valid votes polled by another candidate
= (6000 – 3300) = 2700
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