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# Show that triangle-free planar graphs are four-colorable
Prove that every planar graph without a triangle (that is, a cycle of length three) has a vertex of degree three or less. Then, prove that all planar graphs without triangles are four-colorable without using the four-color theorem.
This is a problem in my textbook that does not have a solution. How to approach this? Induction seems not to work.
• Actually, Grötzsch's Theorem does even better: if $G$ is a planar triangle-free graph then $\chi(G) \le 3$.
– A.S
Aug 2, 2013 at 5:05
For the first part we can use Euler's $F-E+V=2$, where $F$ is the number of faces (counting the face that includes $\infty$), $E$ is the number of edges, and $V$ the number of vertices.
If there are no triangles then each face is enclosed by at least $4$ edges. So $4F/2<E$, i.e. $F<E/2$. From this we get $V>E/2+2$. If all vertices have order $\geq4$ then $4V/2<E$, i.e. $V<E/2$. From this we get $0>2$. So, we cannot have no triangles and at the same time all vertices with degree $\geq4$.
For the second part take a vertex that has degree $\leq3$. Color it and its neighbors with different colors $A,B,C,D$. We can do this only because it has degree $\leq3$. We can delete this vertex and the edges insident on it. If we color the rest of the graph there is always a way to color the deleted vertex. Notice that the graph with this vertex deleted still has no triangles, but less vertices. Apply induction on the number of vertices now.
• There is an error (may be typo) in the first part. It should not be strict inequalities eg: $4F/2\leq E$ and $4V/2\leq E$. Jul 13, 2020 at 2:28
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Latin, the language of the Romans, was originally spoken only in the small region of Central Italy known as Latium. It might well have been displaced by geographically more widespread Italian tongues such as Oscan and Umbrian, but Roman conquests made it the received speech of the peninsula, and ultimately of most wealthy or educated people all over a vast empire.
Writing began in the 7th century BC, when the Latin alphabet was devised as an adaptation of the Greek letters used by the Romans Etruscan neighbours. The earliest known written Latin is on the Lapis Niger, an inscriped block of stone, found in the Forum Romanum, that probably dates from the early 6th century BC. There is no way of establishing how many Romans became literate, but written records, orders and transactions were vital to the efficient running of the Empire, while poems, plays, political and philosophical reflections, letters, prayers and graffiti provided outlets for vivid self-expression that often bring the Romans very close to us.
Commemorative inscriptions and some official decrees appeard on stone or bronze, but a variety of materials were available for other purposes. A wooden tablet with a coating of wax was particularly useful for temporary personal memoranda and school work; the metal or wooden stylus that scratched letters into the wax had a flat top that could be employed for corrections or to erase all the written content so that the tablet could be used again. However, occasion this unstable medium also seems to have served for quite important documents.
Letters and other personal statement could be written with pen and ink on thin sheets of wood. Pens were made of metal, or sharpened reeds of feathers; Roman ink, variously mixing soot, resin and cuttlefish ink, was surprisingly black and durable, as surviving documents have shown. Those who could afford it bought a much less cumbersome, paper-like material made from the papyrus reed (an Egyptian invention from which the word paper is actually derived). This was also used for Roman books, which took the form of long papyrus scrolls. A book, or part-book, was called a volumen, or roll (the origin of the English word volume); the reader held it in both hands, simultaneously rolling and unrolling it so that the text was progressively revealed. Under the Empire, parchment and and vellum (made from hides) were also used as writing surfaces, and the book with pages, or codex, began to be made, but surprisingly it never replaced the much clumsier roll.
Roman interest in books only became intense under the influence of Greek culture in the 2nd century BC, when libraries were part of the booty brought back from the East by Sulla and other successful generals. An influx of well-educated Greek slaves facilitated the development of the book trade in Rome, where something resembling mass production was achieved by booksellers whose slaves wrote out copies of texts dictated by one of their number. The influence of ardent collectors such as Cicero helped to create a fashion for possessing a private library, and booksellers flourished in the capital and ultimately in most parts of the Empire, advertising the latest authors on the pillars outside their shops. The first public library in Rome was founded during the reign of Augustus by the retired politician-poet Gaius Asinius Pollio, and within a century there were over 20, some of which allowed members to take out books for private reading. Authors were paid an outright fee for their works, and even the most famous were unable to live on their literary income. Martial for example, complained that in spite of being read even on the Danube and in Britain, his work brought him little profit. The bookseller-publisher could not afford to be generous, since there was no copyright law and consequently he could never acquire exclusive rights to any publication. As soon as he put a work on sale, a rival bookseller could acquire a copy, put his own scribes to work on it, and publish his own edition. Impecunious authors were therefore dependent on the generosity of one or more patrons and as Martial lamented, there was not a Maecenas in every generation.
Source: Nathaniel Harris : History Of Ancient Rome
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Why should we control the UV doses received? Skin and eyes health risks
Prolonged exposure to ultraviolet radiation can cause significant damage to living organisms:
- The painful experience of a sunburn (erythema) is a visible manifestation of a type of defense mechanism set up by the cells of our skin against excessive UV doses.
- The ultraviolet radiation, and particularly the most aggressive (UV-B), can cause significant damage to the human genome, causing irreversible changes in the gene sequences of our DNA.
Fortunately, in most cases, the organisms are able to overcome these attacks with natural self-correcting mechanisms to control the risks associated with such an exposure. For instance, most of us are able to tan.
However, at higher doses, the human species can:
- develop certain photo-allergies
- undergo accelerated aging of the skin
- suffer from eye cataracts
- develop certain types of skin cancer such as melanoma
Health risks of UV exposure for each human being is not equal
In conclusion, the risks incurred by people who expose themselves to UV, voluntarily or not, are mainly related to:
- type of ultraviolet light absorbed (the intensity of it)
- dose received
- frequency of exposure
- age at which one is exposed (mature skin or not)
Thus, the risks of UV exposure is not the same for each human being. It is a function of skin type or phototype (more or less dark) and there are four different skin types that determine the type of sunscreen you need to apply in each case.
Similarly, plants are also very sensitive to the UV dose absorbed as growth yields slow down with abnormal exposure to UV rays.
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# 1975 USAMO Problems/Problem 3
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
## Problem
If $P(x)$ denotes a polynomial of degree $n$ such that $$P(k)=\frac{k}{k+1}$$ for $k=0,1,2,\ldots,n$, determine $P(n+1)$.
## Solution 1
Let $Q(x) = (x+1)P(x) - x$, and clearly, $Q(x)$ has a degree of $n+1$.
Then, for $k=0,1,2,\ldots,n$, $Q(k) = (k+1)P(k) - k = (k+1)\cdot \dfrac{k}{k+1} - k = 0$.
Thus, $k=0,1,2,\ldots,n$ are the roots of $Q(x)$.
Since these are all $n+1$ of the roots of the $n+1^{\text{th}}$ degree polynomial, by the Factor Theorem, we can write $Q(x)$ as $$Q(x) = c(x)(x-1)(x-2) \cdots (x-n)$$ where $c$ is a constant.
Thus, $$(x+1)P(x) - x = c(x)(x-1)(x-2) \cdots (x-n).$$
We plug in $x = -1$ to cancel the $(x+1)P(x)$ and find $c$:
\begin{align*} -(-1) &= c(-1)(-1-1)(-1-2) \cdots (-1-n) \\ 1 &= c(-1)^{n+1}(1)(2) \cdots (n+1) \\ c &= (-1)^{n+1}\dfrac{1}{(n+1)!} \\ \end{align*}
Finally, plugging in $x = n+1$ to find $P(n+1)$ gives:
\begin{align*} Q(n+1)&=(n+2)P(n+1)-(n+1)\\ (-1)^{n+1}\dfrac{1}{(n+1)!}\cdot(n+1)! &=(n+2)P(n+1)-(n+1)\\ (-1)^{n+1}&=(n+2)P(n+1)-(n+1)\\ (-1)^{n+1}+(n+1)&=(n+2)P(n+1)\\ P(n+1) &= \dfrac{(-1)^{n+1} + (n+1)}{n+2}\\ \end{align*}
If $n$ is even, this simplifies to $P(n+1) = \dfrac{n}{n+2}$. If $n$ is odd, this simplifies to $P(n+1) = 1$. $\Box$
~Edits by BakedPotato66
## Solution 2
It is fairly natural to use Lagrange's Interpolation Formula on this problem:
\begin{align*} P(n+1) &= \sum_{k=0}^n \frac{k}{k+1} \prod_{j \ne k} \frac{n+1-j}{k-j} \\ &= \sum_{k=0}^n \frac{k}{k+1} \cdot \frac{\frac{(n+1)!}{n+1-k}}{k(k-1)(k-2) \dots 1\cdot (-1)(-2) \dots (k-n)} \\ &= \sum_{k=0}^n \frac{k}{k+1} (-1)^{n-k}\cdot \frac{(n+1)!}{k!(n+1-k)!} \\ &= \sum_{k=0}^n (-1)^{n-k} \binom{n+1}{k} - \sum_{k=0}^n \frac{(n+1)!(-1)^{n-k}}{(k+1)!(n+1-k)!} \\ &= -\left(\sum_{k=0}^{n+1} (-1)^{n+1-k} \binom{n+1}{k} - 1\right) + \frac{1}{n+2} \cdot \sum_{k=0}^n (-1)^{n+1-k} \binom{n+2}{k+1} \\ &= 1 + \frac{1}{n+2} \left(\sum_{k=-1}^{n+1} (-1)^{n+2 - (k+1)} \binom{n+2}{k+1} - (-1)^{n+2} - 1\right) \\ &= \boxed{1 - \frac{(-1)^n + 1}{n+2}} \end{align*} through usage of the Binomial Theorem. $\square$
~lpieleanu (minor editing and reformatting)
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# Thread: Point on a line closest to the origin
1. ## Point on a line closest to the origin
On my last calc test, they asked f(x)=10x+5, what point in the function is closest to the origin?
I solved it by finding the normal line which intersects the origin, then finding where that line intersects f(x). (We haven't gotten the tests back yet, so I don't know how if that worked)
Anyway, this didn't involve any calculus, so I'm concerned that the problem was written wrong, and was supposed to be something like $10x^{3}+5$.
This question, I am not sure how to solve, so can anyone help me figure out how I would find the point on the function $f(x)=10x^{3}+5$ which is closest to the origin?
And if this method does not require normal lines, could you also explain to me how I would find the normal line for such a function? (I have a theory that it would be $-\frac{1}{30x^{2}}$ but that's just a theory.)
2. Originally Posted by angel.white
On my last calc test, they asked f(x)=10x+5, what point in the function is closest to the origin?
I solved it by finding the normal line which intersects the origin, then finding where that line intersects f(x). (We haven't gotten the tests back yet, so I don't know how if that worked)
Anyway, this didn't involve any calculus, so I'm concerned that the problem was written wrong, and was supposed to be something like $10x^{3}+5$.
This question, I am not sure how to solve, so can anyone help me figure out how I would find the point on the function $f(x)=10x^{3}+5$ which is closest to the origin?
And if this method does not require normal lines, could you also explain to me how I would find the normal line for such a function? (I have a theory that it would be $-\frac{1}{30x^{2}}$ but that's just a theory.)
this is an optimization problem. what you are really being asked to do is to minimize the distance function between the two points (0,0) and (x, 10x + 5)
3. Originally Posted by Jhevon
this is an optimization problem. what you are really being asked to do is to minimize the distance function between the two points (0,0) and (x, 10x + 5)
My solution I used on the test was that the normal line which intersects (0,0) is $y=-\frac{1}{10}x$ Hmm, well I got the same answer either way, so hopefully my instructor won't take off points for not using calculus to solve it :/
Anyway, how would I find the normal line for:
$f(x)=10x^{3}+5$?
4. Originally Posted by angel.white
Hmm, perhaps you can help me figure out where I am going wrong.
My solution I used on the test was that the normal line which intersects (0,0) is $y=-\frac{1}{10}x$ Hmm, well I got the same answer either way, so hopefully my instructor won't take off points for not using calculus to solve it :/
Anyway, how would I find the normal line for:
$f(x)=10x^{3}+5$?
well, we know that $f'(x) = 30x^2$ gives the slope for the tangent line.
this means that the slope for the normal line is given by $- \frac 1{30x^2}$ as your theory was (the normal line is a line perpendicular to the tangent line, and so their slopes are the negative inverses of each other).
so now the challenge is to find the line with that slope going through the origin to our curve. which shouldn't be too hard i think
5. Originally Posted by Jhevon
well, we know that $f'(x) = 30x^2$ gives the slope for the tangent line.
this means that the slope for the normal line is given by $- \frac 1{30x^2}$ as your theory was (the normal line is a line perpendicular to the tangent line, and so their slopes are the negative inverses of each other).
so now the challenge is to find the line with that slope going through the origin to our curve. which shouldn't be too hard i think
Hmm, I'm actually not sure how to do that, it has a vertical asymptote at x=0, so moving it up and down won't help, and I can't figure out how to move it left/right without changing the slope.
However, your suggestion earlier will obviously work to figure out the closest distance between the origin and the line, so if he pops this question on a quiz, I'll be fine, won't need the normal line to solve it.
6. Originally Posted by angel.white
Hmm, I'm actually not sure how to do that, it has a vertical asymptote at x=0, so moving it up and down won't help, and I can't figure out how to move it left/right without changing the slope.
However, your suggestion earlier will obviously work to figure out the closest distance between the origin and the line, so if he pops this question on a quiz, I'll be fine, won't need the normal line to solve it.
yeah, go the optimization route. i think i see a way to solve it using the normal line, but i end up with a quartic, which is too much trouble to go through for something like this
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## K-2 Students Be Shoppin’!!!!
Anyone who has seen Happy Numbers in action can tell that students love it. But what do teachers think? One compliment we often receive is that Happy Numbers offers a diverse range of activities. By presenting the same concept using a variety of tools, students are given the opportunity to investigate concepts thoroughly. Another is that the math itself is what piques student interest, rather than a separate, unrelated game thrown in as a reward for enduring a set of dry math problems.
These are two great reasons to use shopping-based exercises throughout your math instruction. At Happy Numbers, we don’t treat shopping as a separate topic, but strategically integrate it throughout our curriculum to target key skills. For example, students don’t jump right from combining coins to making change. Instead, each of these skills is presented sequentially along with similar skills that use different representations (e.g., base-10 blocks, equations, hundred chart). This way, the shopping exercises relate to student learning and students are equipped with the skills they need to succeed.
To learn more about using multiple representations to build deep conceptual understanding, see our posts about base-10 blocks, number line, pan balance, and hundred chart on the Happy Numbers blog.
Shopping is an authentic, mathematical activity that calls on a wide range of skills. It’s also fun! Even if you’re not using Happy Numbers, this post will give you ideas about how to scaffold shopping exercises to meet students’ needs at various levels. You can re-create shopping exercises in the classroom using plastic coins and images with prices written on them. If your class does use Happy Numbers, of course the scaffolding is built right into our program and students will even receive immediate feedback and support as they work through each activity.
Here’s how we use shopping activities to scaffold math skills…
All of the exercises mentioned here are part of the HappyNumbers.com course and are presented along with exercises using other representations.
## 1. Begin with the simplest addition skills.
As soon as students are able to add two numbers on their own, we present them with the first shopping exercise. Here, students choose two numbers (not coins) out of three to reach a given sum:
All sums are 10 or lower, and students have already worked extensively adding small numbers using objects, the number line, and equations. Now they can enjoy practicing their new skills with number bonds while “buying” treats.
## 2. Increase the challenge to more complex addition.
Next, we keep the shopping experience the same, but present students with more challenging addition in two ways. The first challenge is to simply purchase items with higher prices (up to 20):
The next challenge at this step is to use three addends to reach a given sum (again, up to 20):
As you can see, these first three shopping exercise increase in difficulty quite a bit, which is why we don’t recommend presenting them to students in immediate succession. There is quite a bit of math that goes on between each of these exercises in order to prepare students to succeed with each task.
## 3. Use coins to reach a given sum.
Once students are familiar with adding numbers to reach a sum, we introduce the constraint of using only certain “coins”. (We use a more universal labeled coin to focus on numeracy rather than coin recognition.) We begin with just 10s and 1s – a structure familiar to students from their work with base-10 blocks and the hundred chart. At this point, we also increase the total sum to accommodate the use of more tens:
After students demonstrate mastery with 10s and 1s, we introduce 50s and 5s to achieve the same task:
Here, it is interesting for teachers to observe students’ strategies. Will they continue adding 10+10+10+10+10+10 or will they discover the “shortcut” of using 50+10? What will they do if they run out of 1s? To ensure students try both options, we eventually limit the number of certain coins. This way, students realize they don’t have enough and must use a higher denomination.
## 4. Count a collection of given coins.
As students progress in this topic, we reverse their role – instead of choosing coins to reach a given total, they now have a given set of coins and must determine the total. In essence, they’ve gone from the role of customer to that of cashier.
Again, we begin the exercise by using only 10s and 1s:
Then we increase the complexity by including 5s:
The final challenge at this step is to count a given collection of coins and then compare that with several given totals. Students love figuring out which toy they can afford! This exercise is more open-ended (just like real shopping) because there are several possible correct and incorrect responses.
At this point, if a student selects a toy that is lower than the given amount, we recommend they do not yet calculate the change. This is a more advanced two-digit subtraction skill that we’ll address later on. On Happy Numbers, students are shown the coins that make up their change, but they are not asked to count them or calculate it.
## 5. Find the total of multiple items.
Moving toward increasingly authentic shopping experiences, students next use coins to pay for two items:
They have been well-versed in two-digit addition up to this point, so they have multiple strategies available to them in solving each problem:
– Add the two numbers in their head and pay for the total
– Pay for one item, then the next item without totaling
– Pay for all the 10s first, then all the 1s
Since the second strategy doesn’t require mental addition, we don’t let students rely on it too long. Again, we limit their coin selection so they must find the total instead of using the “pay for one then pay for another” trick:
Again, our emphasis is on number values rather than actual coin recognition, which is why we’ve chosen to include 2s here and not 25s.
Next, we ensure students use mental two-digit addition by removing the coins and focusing just on the two items and the total. You can present this task as one in which students choose two items and find the total or, as we’ve done, give students the total and have them determine the two items that were purchased. The latter approach, in which they must figure out what the cat bought, seems to really hold their interest:
## 6. Complete the entire shopping process.
Finally, students are well-prepared to take on every step in an authentic shopping experience:
a) Count your money and choose items to purchase.
b) Present the items to a cashier and choose coins with which to pay.
c) Determine the amount of change owed and count it out. (Too bad this cashier didn’t use Happy Numbers or he could do it himself!)
d) Enjoy the rewards of your successful shopping trip!
As you can see, these shopping exercises increase in difficulty quite a bit, which is why we spread them across grade levels and lessons rather than presenting them in immediate succession. Happy Numbers carefully integrates all of these steps, and the prerequisite skills needed, to give students deep conceptual understanding of math.
We’ve found that shopping exercises give students great practice with meaning. We hope you’ll give them a try in your classroom. Kids love them and you’ll love them, too!
Educationally yours,
Evgeny & Happy Numbers Team
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Matthias, 1557–1619, Holy Roman emperor (1612–19), king of Bohemia (1611–17) and of Hungary (1608–18), son of Holy Roman Emperor Maximilian II. He was appointed governor of Austria (1593) by his brother, Holy Roman Emperor Rudolf II. He formed a close association there with the bishop of Vienna, Melchior Klesl, who later became his chief adviser. In 1605, Matthias forced the ailing emperor to allow him to deal with the Hungarian Protestant rebels. The result was the Peace of Vienna (1606), which guaranteed religious freedom in Hungary. In the same year Matthias was recognized as head of the house of Hapsburg and as future Holy Roman emperor, as a result of Rudolf's illness. Allying himself with the estates of Hungary, Austria, and Moravia, Matthias forced (1608) his brother to yield rule of these lands to him; Rudolf later ceded (1611) Bohemia. After Matthias's accession as Holy Roman emperor, his policy was dominated by Klesl, who hoped to bring about a compromise between Catholic and Protestant states within the empire in order to strengthen it. Matthias had already been forced to grant religious concessions to Protestants in Austria and Moravia, as well as in Hungary, when he had allied with them against Rudolf. His conciliatory policies were opposed by the more intransigent Catholic Hapsburgs, particularly Matthias's brother Archduke Maximilian, who hoped to secure the succession for the inflexible Catholic archduke Ferdinand (later Holy Roman Emperor Ferdinand II). The start of the Bohemian Protestant revolt in 1618 provoked Maximilian to imprison Klesl and revise his policies. Matthias, old and ailing, was unable to prevent a takeover by Maximilian's faction. Ferdinand, who had already been crowned king of Hungary (1617) and of Bohemia (1618), succeeded Matthias as Holy Roman emperor.
The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.
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It seems everyday more and more information is being uncovered about trees and the many mysteries within them. We know that they are alive, but it seems they are even more alive than we may have thought.
Trees are interconnected underground, we also now know that trees can communicate with one another, but recently scientists have discovered that trees actually have a sort of heartbeat, it is just so slow that they’ve never noticed before.
Up until recently scientists had thought that water moved through trees by the process of osmosis, in a sort of continuous matter, but now they’ve discovered that the trunks and branches of the trees are actually contracting and expanding and essentially pumping water up from the roots to the leaves, kind of like how our heart pumps blood throughout our bodies.
Unlike our bodies and our pulse however, a tree’s is much slower and beats only once every two hours or so and instead of regulating blood pressure it actually regulates the water pressure flowing through the tree.
“We’ve discovered that most trees have regular periodic changes in shape, synchronized across the whole plant, which imply periodic changes in water pressure.”
How Did They Find This Out?
A study conducted in 2017 Zlinsky and his colleague, Anders Barfod, used terrestrial laser scanning to monitor 22 different tree species in an effort to observe the shape of their canopies and how they changed.
All measurements were taken in greenhouses at night in order to rule out the sun and also the wind as factors in the movements in the trees. In several of the trees it was observed that branches moved up and down about a centimeter every couple of hours.
After finishing up the night study the researchers came up with a theory about what they believe that movement represents. They believe that the motion is a signal that the trees are pumping water up from their roots and distributing it through their branches.
The researchers believe that this discovery indicates that trees might actually have a type of heartbeat.
“In classical plant physiology, most transport processes are explained as constant flows with negligible fluctuation in time,” Zlinszky told New Scientist. “No fluctuations with periods shorter than 24 hours are assumed or explained by current models.”
Unfortunately at this time there is still no explanation as to how the pumping action works. They speculate that perhaps the trunk squeezes the water pushing it up through the xylem, which is a system of tissue in the trunk that transports water and nutrients to all the branches and the leaves.
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Commonly found in pools with rock or silt substrate bottoms. These fish also prefer swift currents and runs of small to medium rivers, and are often present around fallen logs and debris.
Edward D. Cope first described the species now known as the robust redhorse in North Carolina’s Yadkin River in 1869. The historic range of the species was from the Altamaha River in Georgia to North Carolina’s Pee Dee River. Limited wild populations exist today in the Ocmulgee and Oconee Rivers in Georgia, as well as the Savannah River in Georgia and South Carolina, and the Pee Dee River in North Carolina.
Robust redhorses eat small crustaceans, mollusks, insects, algae, and detritus (pieces of dead stuff).
The discovery of the robust redhorse in 1869 by Edward D. Cope was misinterpreted by fish biologists later on. Specimens of robust redhorse were mistakenly identified as smallfin redhorse.
For over 100 years the redhorse went unrecognized, lost in the mistake made by scientists long ago. Between 1980 and 1992, however, large specimens of true robust redhorse were discovered in rivers of the Carolinas and Georgia. These latter fish were correctly identified.
The habitat and life history of the species remains a mystery. The fish has been difficult to study because of the effort needed for sampling and their close relation to other species.
Like all fish declared suckers, the robust redhorse has a bottom facing mouth that is used to feed on bottom dwelling animals. This mouth is made up of two fleshy lips that are surrounded by finger-like projections called “papillae.” These lips allow this fish to create enough suction to remove algae from rocks and feed on crustaceans, mollusks, insects and detritus. These bottom feeding fish provide an important ecological service by “cleaning up” the bottoms of aquatic habitats of algae and dead materials. This service improves the overall health of habitats that are important for a variety of species.
Aside from being “lost” to science for 100 years, the robust redhorse has faced significant reduction in population due to habitat degradation. Agricultural, residential, and commercial development along robust redhorse habitat has lead to siltation of spawning grounds; stream alteration and channeling has caused a decline in robust redhorse habitat. In addition, the non-native blue and flathead catfish are known to predate on robust redhorses, which have a low number of individuals reaching sexual maturity. Lastly, dams and weirs block historic migration routes of reproducing fish. An inter-agency alliance called the Robust Redhorse Conservation Committee is working to restore habitat for this species, and replenish rivers where they were once abundant. What’s more, awareness of the specie’s decline has lead to research for determining life history, population dynamics, and genetics. Through education, RRCC and institutions such as the South Carolina Aquarium show the need for habitat protection and stronger conservation ethics.
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# Exercise C.1.4
In how many ways can we choose three distinct numbers from the set $\{1,2,\ldots,99\}$ so that their sum is even?
There are 49 even numbers and 50 odd in that set. To get an even number, we either need two odd and one even or three even.
$$\langle \text{2 odd, 1 even} \rangle = 49\frac{50!}{2! \cdot 48!} = \frac{50 \cdot 49^2}{2} \\ \langle \text{3 even} \rangle = \frac{49!}{3! \cdot 46!} = \frac{49 \cdot 48 \cdot 47}{6} \\ \langle \text{even sum} \rangle = \frac{49 \cdot 48 \cdot 47}{6} + \frac{50 \cdot 49^2}{2} = 78449$$
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The B-24: The Great Liberator
On February 24, 1943, three squadrons of B-24 Liberators—goliath, four-engine, 56,000-pound bombers—streaked toward Germany to strike Hitler’s vaunted Luftwaffe at its heart, targeting a key production facility in the town of Gotha, Germany.
Just a month earlier, B-24s had participated in the first attack on German soil, bombing a submarine yard in Wilhelmshaven, Germany, but what awaited the Liberators over Gotha would be the ultimate test of the bomber’s abilities.
Eighty minutes into their flight, German fighters swooped in, taking their toll on the B-24 squadrons. Then came a firestorm of antiaircraft cannon shells, rockets, and air-burst bombs, turning the skies into a hellish expanse of bullets, smoke, and flak.
Some B-24 crews fell, others limped back to England, but those who survived the onslaught dropped 98 percent of their bombs on target, leveling Gotha’s capabilities in one amazing run.
Considered one of the best examples of precision bombing of the war, the raid on Gotha devastated German aircraft production and established the B-24 as one of the Allies’ most trusted bombers.
Turning the Tide in Europe
Conceived in 1938 by Consolidated Aircraft, a Lockheed Martin legacy company, the original B-24 prototype was designed to fly faster and carry a larger payload than the US Army Air Corps’s B-17 Flying Fortress. In time the B-24 would boast a long, tapered wing atop its fuselage, which allowed impressive long-range cruising capabilities. A B-24 could reach 290 miles per hour and carry a 5,000-pound bomb load for 1,700 miles, giving it a longer range, greater speed, and a bigger payload than its B-17 cousin.
By 1941, B-24s were being shipped to Great Britain, where they were given the name Liberator and adapted for a variety of purposes, including coastal patrol, protecting critical Atlantic cargo ship convoy crossings. The Liberators’ range proved invaluable in scouting and destroying German U-boats, creating safe passage for Allied transports and destroyers across Europe. They also bombed German oil refineries and attacked critical targets in Italy, changing the tide of the Allies’ Mediterranean campaign.
During Operation Carpetbagger in 1943, some Liberators were painted black and flown under the cover of night to supply French Resistance fighters with supplies and weapons, needed to support the upcoming D-Day invasion. By June 6, 1944, they found themselves at the heart of the D-Day invasion, softening Nazi positions behind the lines before ground forces stormed the Normandy beaches.
Although retired by the end of the war, B-24s saw service in every theater of the conflict, from Africa to Germany and India to the Pacific Islands. In total, a stunning 18,482 B-24s were produced to wage war against the Axis powers. No other American combat aircraft in history was produced on a larger scale.
Sources and Additional Reading
- Boyne, Walter J. Air Warfare: An International Encyclopedia: A-L. Santa Barbara, Calf.: ABC-CLIO, 2002.
- Bruning, John R. Bombs Away! The World War II Bombing Campaigns Over Europe. Minneapolis: Zenith Imprint, 2011.
- Military Factory. “Consolidated B-24 Liberator Medium/Heavy Bomber.” http://www.militaryfactory.com/aircraft/detail.asp?aircraft_id=80.
- NASA. “Quest for Performance: The Evolution of Modern Aircraft.” http://www.hq.nasa.gov/pao/History/SP-468/ch5-3.htm.
- National Air Force Museum. “Operation Carpetbagger.” http://www.nationalmuseum.af.mil/factsheets/factsheet.asp?id=1502.
- Stout, Jay. Fortress Ploesti: The Campaign to Destroy Hitler’s Oil Supply. Havertown, PA: Casemate Publishers, 2011.
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In Ragtime, Doctorow often makes allusions to familiar historical figures and events. However, he often alters certain details or entirely fabricates circumstances. In this way, the novel adopts an element of fantasy as well as addressing the subjectivity of historical accounts. Doctorow rejects one-sided absolutes in favor of a more complex view of history enriched by a multiplicity of voices. To this end, Doctorow's many interconnected characters and events draw attention to individuals' various reactions to similar events and circumstances. Through this method of characterization, the reader gains more profound insight into both the character himself and the broader social trends implicit in the character's reactions.
Imagery plays an important role in this novel. The motion picture, an innovation of the Progressive Era, gains prominence during this time, with the threat it presented to traditional art and culture, and the relatively inexpensive cost of attending a film. Tateh achieves relative well-being through his involvement in the production of movies. In addition, Doctorow's interest in imagery manifests itself stylistically in his writing. The novel also expresses an interest in the increased use of duplication as a result of technological advancements, and the consequent loss of a sense of individuation.
In Ragtime, E.L. Doctorow employs a unique narrative style. The narrator seems to be neither an omniscient and uninvolved individual nor any one specific character. Critics have varying opinions on the origin of the narrative voice; most critics agree that the voice appears to be that of an American writing in 1974. The narrator's sense of historical perspective, as well as his use of ironic and rhetorical commentary, seems to support this notion. The narrator's knowledge about the little boy's thoughts and feelings might lead the reader to believe the little boy narrates the story; however, the narrative voice remains in the third person. In addition, perhaps Tateh's little girl provides the narrative voice. Another possibility lies in the notion that the little girl and the little boy narrate the story together. Tateh, Mameh, and the little girl seem to find their parallel in Father, Mother, and the little boy; perhaps each child provides different elements to the narration and to the story line, to produce a more comprehensive image of America at the turn of the century. The recurrent presence of "we" throughout the novel supports this belief that the two voices narrate together.
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Moving around the cave
Since cave diving is different from other recreational diving activities, many of the techniques people use are also much different. Divers are taught to swim in a prone, or face down, position, with the knees bent and the fins elevated above the plane of the body. This is mainly a precaution against kicking the bottom of a cave and stirring up sediment, but it also offers a good streamline and creates little resistance to the water.
Cave divers move about a cave by using a simple technique called "pull and glide" -- using the tips of their fingers, divers look for crevices in rock for a place to hook onto. The rock is usually something hard and porous like limestone, so it should have lots of pockets and places to grab. After grabbing hold, divers pull and release, gliding through the cave with relative ease.
Cave divers learn how to use mostly their feet for directional changes along with short flutter kicks, and, in the case of solid limestone, some can push off a cave ceiling with their feet to propel themselves along.
Divers can also take along battery-powered diver propulsion vehicles (DPVs) to make swimming easier. Although there are many different types, tow-behind DPVs are the most common, which pull divers through caves. DVPs help divers use less oxygen since they're not exerting themselves as much, and they can significantly increase the length of a dive.
Because there is little to no visibility in caves and cave divers must use their own source of light, guidelines must be placed to ensure people can find their way back to a cave's entrance.
Most caves already have guidelines in place from past explorers -- these are called "gold lines" because of their yellowish color. They consist of braided nylon string and are usually a bit smaller in diameter than regular rope at about an eighth of an inch. These are placed throughout the main tunnels of a cave. Labyrinthine caves also have smaller side tunnels, and these are provided with smaller, white lines. They don't contact the main line; instead, they usually end within 5 to 10 feet of the main line.
The main line of a cave does not extend to the exit -- this prevents open-water divers or untrained or uncertified people from viewing it as an invitation to enter the cave. Therefore, a main guideline typically starts 50 to 100 feet inside a cave.
Still, it's a cave diver's responsibility to run a temporary line, or entry line, along a reel from the outside of the cave in order to maintain a continuous guideline from open-water to the main line. This provides direct access to a cave's exit. To make an entry line, divers make an initial tie-off to something sturdy, like a big rock. A secondary tie-off is also made in case the first one comes loose. The diver must be able to swim along the line with his hand around it, making an "OK" sign, and with his eyes closed make his way out of the cave. The line shouldn't be run near obstructions in order to avoid snags and keep out of the way of other divers.
Dorf markers, or small, plastic directional arrows, can be tied to lines. These point toward exits, just in case a diver becomes disoriented. Clips, markers that resemble clothespins, are also used at points for notation reasons, including max penetration (the furthest point reached inside the cave) and points of interest for other divers.
The average cave dive will last in excess of one hour, but some can last for as long as 15 hours if the right equipment and gas supply is available. Divers generally use what's called the "rule of thirds" -- when one third of a diver's air supply is gone, he will stop the dive and begin moving toward the cave's entrance.
To learn more about training and certification for cave divers, read the next page.
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# How do you determine the amplitude, period, and shifts to graph y=2cosx?
Jul 12, 2017
$\text{see explanation}$
#### Explanation:
$\text{the standard form of the cosine function is}$
$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a \cos \left(b x + c\right) + d} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
$\text{where amplitude "=|a|," period } = \frac{2 \pi}{b}$
$\text{phase shift " =-c/b," vertical shift } = d$
$\text{here } a = 2 , b = 1 , c = d = 0$
$\Rightarrow \text{amplitude "=|2|=2," period } = 2 \pi$
$\text{there are no shifts}$
graph{2cosx [-10, 10, -5, 5]}
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Pre-school Theme Day
Letter “J” Day
To help my 3 year old son learn that the alphabet was more than just a song, we had letter days. Having them individually helped him to recognize that each letter is different just like each shape is different. After our theme day we’d review that letter for the week until the next theme day.
Print out the Family Theme Day Planner and decide which activities you’d like to do and in what order.
The obvious choice for letter days is the “ABCD...” alphabet song - http://www.kididdles.com/lyrics/a004.html
Many different Children’s Music Recordings have other alphabet songs (like Sharon, Lois and Bram), check your children’s collections to see what you have on hand.
There are songs that emphasize the sound of each letter, too (One sounds like “Farmer in the Dell” but says “J says jah, J says jah, every letter makes a sound, J says jah”).
You can find many free colouring pages online by using your favourite search engine and typing in “Alphabet Coloring Pages ” (you can often find alphabet pages with favourite characters on them too like the Sesame Street Characters) or print out my “Big J Little j” Colouring Page.
While colouring the page, emphasize the shape by helping your child trace it with his/her finger and emphasize the sound (for Letter J day emphasize the sound “ja” ).
Raid your child’s bookshelves to find any alphabet books.
Go to the library with your child to find some alphabet books.
Go to the library on your own to find alphabet books to have already on hand for your theme day. Many libraries allow you to go online and search for titles based on subject (search for “alphabet” under “Children’s Books”). Reserve them if you can to save time.
Try to find some of these fun alphabet books:
· The A to Z Beastly Jamboree, by Robert Bender, Lodestar Books, 1996 – Each page of this alphabet book has a humorous picture of an animal doing an action that starts with the same letter.
· The Jazzy Alphabet, by Sherry Shahan and illustrated by Mary Thelen, Philomel Books, 2002 – This brightly illustrated book has jazzy fun while looking at each letter of the alphabet.
· Jeepers Creepers: A Monstrous ABC, by Laura Leuck and illustrated by David Parkins, Chronicle Books, 2003 – Twenty-six monsters, each with a name that starts with each letter of the alphabet, go to school and are surprised by what they see in their “Creepy Creatures ABC.”
· J is for Jamaica, by Benjamin Zephaniah and photographs by Prodeepta Das, Frances Lincoln Children’s Books, 2006 – With photographs from Jamaica and rhyming text this alphabet book looks at things unique to the island country like ackee fruit, cricket, Kingston etc..
· Journey Around New York from A to Z, by Martha and Heather Zschock, Commonwealth Editions, 2002 – This informative alphabet book offers a brief alliterative text in a framed picture and then under a larger one gives the reader some New York history.
· Jousting with Jesters: An ABC for the Younger Dragon, by Martin Springett, This fun alphabet book has a little dragon who grows with each page on a quest to find his flame and on his journey there are many things he must do (all beginning with each letter of the alphabet in fun phrases of alliteration). There are also many hidden things that start with each letter in the illustrations, making it a fun search and find book, as well.
Letter J Collage:
Materials: A copy of my Jj worksheet, old magazines, child-safe scissors, washable glue stick, damp facecloth for sticky fingers.
Step 1: Look through old magazines with your child and together look for things that start with the letter J.
Step 2: Help your child cut out the letter H pictures from the magazine to make a pile of pictures to glue on the Jj worksheet.
Step 3: Show your child how to glue the pictures onto the collage and then let him/her glue the pictures on the paper however he/she likes.
Step 4: When the collage is dry display (fridge, bulletin board, child’s door) or glue into Family Theme Scrapbook.
Letter J Sticker Collage:
Materials: Coloured paper, stickers of things that start with the letter J (or of the letter J if you can find some alphabet stickers).
Step 1: Have your child pick the colour of paper to use for the background
Step 2: Give your child the stickers and let him/her stick them to the coloured paper however he/she wants.
Jar Rim Stamps:
Materials: empty jars of different sizes, paints, paint brush, a copy of my Jj worksheet , cloth for messy fingers, art smock or old shirt to wear, newspaper or plastic to cover the work space .
Step 1: Have your child pick the colour(s) of paint he/she wants to use.
Step 2: Remind your child that the word jar starts with the “ja” sound.
Step 3: Let your child draw all over the Jj page with washable markers.
Step 4: Show your chid how to take a paint brush and coat the jar rim with paint and then press it on the worksheet to leave a circle.
Step 5: Let it dry and then display or glue in your Family Theme Day Scrapbook.
Serve some jelly or jam on toast or crackers for a Letter J snack.
For something different snack on some beef jerky for this theme day.
Any type of Juice would be the perfect drink for this theme day.
For a sweet treat try some ju-jube candies.
If you can find some make a sandwich with Jarlsberg cheese.
Make some Japanese cuisine (a stir-fry or noodle soup, or sushi-maki rolls from the grocery deil would be the easiest) or go to a restaurant for a treat for a Letter “J” dinner.
Make some jambalaya (search your cookbooks or look online for a recipe).
Make some jerk chicken using jerk seasoning for some spicy poultry.
Make some Jello in the afternoon for a jiggly J dessert.
Materials: A copy of my Alphabet Chart (from Letter A day displayed on your fridge or on a bulletin board), a copy of the Cut-out JjCard, markers or crayons, child-safe scissors, glue-stick, face cloth for sticky fingers.
· Step 1: Lead your child to the Alphabet Chart on your fridge, bulletin board or taped to a wall and review the Letters A to I
· Step 2: Have your child colour the Cut-out Jj Card.
· Step 3: Help your child cut the letter Jj card out.
· Step 4: Have your child apply glue to the back of the Jj card and glue it on the Alphabet Chart (or you can have your child use tape) in the appropriate spot.
· Step 5: Review what letter it is and what sound it is throughout the week by pointing to the chart.
Fill a pie plate with sand, sugar or salt and teach your child how to trace the letter J in the sand. When you are finished tracing dump the sand in a re-sealable bag to use on another day.
Review the entire alphabet by using a set of flash cards (found at book stores, educational stores, even craft stores) or make your own by writing each letter on an individual index card.
There are many different websites that offer games for preschoolers. You can find them by looking up your child’s favourite television characters. Here are two from the Sesame Street website:
Big Bird’s Letters is a simple game because it only involves your child pressing any letter on the keyboard and then the letter appears along with a picture that starts with that letter:
Letters to Big Bird is another alphabet game to play together. In this game Big Bird literally receives a letter in his mail box and your child has to click on something on his shelf that starts with that letter:
Play “I Spy With My Little Eye” only trying to find things that start with the letter J.
This is similar to “I Spy” in that you walk around your neighbourhood and try to find things that start with different letters of the alphabet. For Letter I Day find things that start with J. You can also do this while in a car or bus etc..
If you have an alphabet puzzle this theme day is the perfect time to play with it together.
Play with any other educational toys that focus on the alphabet.
Search through your child’s DVD/ video collection (or visit your local library before hand or the Video Store) to find your child’s favourite shows with a focus on teaching the alphabet.
Try to find these titles:
· Blue’s Room: Alphabet Power, Viacom International Inc., 2005 – This DVD has two episodes of blue’s Room and two of Blue’s Clues. The first two shows deal with the alphabet and the last two more with writing and reading.
· Pocket Snails: Letter Adventure, Soaring Star Productions, 2004 – These two simple shows are about three snails who live in a little boy’s pocket who help him learn the alphabet by taking photos of them in Letter Land. One show highlights the Upper Case letters and the other is identical except it showcases the Lower Case letter. There is no focus on the phonetic sounds of the alphabet in these shows but the repetition makes it a good show to reinforce letter recognition.
· Rock n’ Learn: Alphabet Exercise, Rock ‘N Learn, Inc., 2005 – this show has a song for every letter of the alphabet that also incorporates movement like S for Spin and T for Twist.
· Sesame Street: All-Start Alphabet – There’s So Much to See Between A and Z!, Sesame Workshop, 2005 – This fun DVD has capital A and Z interviewing people at a mall about the alphabet while also highlighting each letter with individual skits from the show Sesame Street. Adults might enjoy it because it includes segments with Sheryl Crow, Norah Jones, and the Dixie Chicks to name a few of the celebrities featured.
· Sesame Street: Learning About Letters, Children’s Television Workshop, 1986 – This is a great video using classic clips (that I remember as a child) throughout as Big Bird and friends search for things that start with each letter of the alphabet.
· Sharon, Lois & Bram ABC’s: Alphabet sing & dance-along, elephant Records, 2003 – this one reviews the alphabet using different songs about things that start with different letters.
J is for jellyfish
Letter “J” Collage
Photo: C Wright
Letter “J” Sticker Collage
Jar Rim Stamps
Letter tracing in sugar
J is for Joshua Tree
Photo: C Wright
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To help answer one of the great existential questions – how did life begin? – a new study combines biological and cosmological models. Professor Tomonori Totani from the Department of Astronomy looked at how life’s building blocks could spontaneously form in the universe – a process known as abiogenesis.
If there’s one thing in the universe that is certain, it’s that life exists. It must have begun at some point in time, somewhere. But despite all we know from biology and physics, the exact details about how and when life began, and also whether it began elsewhere, are largely speculative. This enticing omission from our collective knowledge has set many curious scientists on a journey to uncover some new detail which might shed light on existence itself.
As the only life we know of is based on Earth, studies on life’s origins are limited to the specific conditions we find here. Therefore, most research in this area looks at the most basic components common to all known living things: ribonucleic acid, or RNA. This is a far simpler and more essential molecule than the more famous deoxyribonucleic acid, or DNA, that defines how we are put together. But RNA is still orders of magnitude more complex than the kinds of chemicals one tends to find floating around in space or stuck to the face of a lifeless planet.
RNA is a polymer, meaning it is made of chemical chains, in this case known as nucleotides. Researchers in this field have reason to believe that RNA no less than 40 to 100 nucleotides long is necessary for the self-replicating behavior required for life to exist. Given sufficient time, nucleotides can spontaneously connect to form RNA given the right chemical conditions. But current estimates suggest that magic number of 40 to 100 nucleotides should not have been possible in the volume of space we consider the observable universe.
“However, there is more to the universe than the observable,” said Totani. “In contemporary cosmology, it is agreed the universe underwent a period of rapid inflation producing a vast region of expansion beyond the horizon of what we can directly observe. Factoring this greater volume into models of abiogenesis hugely increases the chances of life occuring.”
Indeed, the observable universe contains about 10 sextillion (10^22) stars. Statistically speaking, the matter in such a volume should only be able to produce RNA of about 20 nucleotides. But it’s calculated that, thanks to rapid inflation, the universe may contain more than 1 googol (10^100) stars, and if this is the case then more complex, life-sustaining RNA structures are more than just probable, they’re practically inevitable.
“Like many in this field of research, I am driven by curiosity and by big questions,” said Totani. “Combining my recent investigation into RNA chemistry with my long history of cosmology leads me to realize there is a plausible way the universe must have gone from an abiotic (lifeless) state to a biotic one. It’s an exciting thought and I hope research can build on this to uncover the origins of life.”
IMAGE SOURCE: Creative Commons
Words matter. Images matter. The Scientific Inquirer needs your support. Help us pay our contributors for their hard work. Visit our Patreon page and discover ways that you can make a difference. http://bit.ly/2jjiagi
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## Precalculus (6th Edition) Blitzer
The inverse matrix of the provided matrix is ${{A}^{-1}}=\left[ \begin{matrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{6} & 0 \\ 0 & 0 & \frac{1}{9} \\ \end{matrix} \right]$.
Consider the provided matrix $A=\left[ \begin{matrix} 3 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 9 \\ \end{matrix} \right]$. Compute matrix in the form of: $\left[ \left. A \right|I \right]$ Augment matrix with identity matrix is: $\left[ \left. A \right|I \right]=\left[ \begin{matrix} 3 & 0 & 0 & 1 & 0 & 0 \\ 0 & 6 & 0 & 0 & 1 & 0 \\ 0 & 0 & 9 & 0 & 0 & 1 \\ \end{matrix} \right]$ Now, we will use row operations to reduce in row echelon form for the inverse: \begin{align} & {{R}_{3}}\to \frac{1}{9}\times {{R}_{3}}, \\ & {{R}_{2}}\to \frac{1}{6}\times {{R}_{_{2}}}, \\ & {{R}_{1}}\to \frac{1}{3}\times {{R}_{1}} \\ \end{align} The resulting matrix is: \begin{align} & \left[ \left. A \right|I \right]=\left[ \begin{matrix} 1 & 0 & 0 & \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 & 0 & \frac{1}{6} & 0 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{9} \\ \end{matrix} \right] \\ & =\left[ \left. I \right|B \right] \end{align} Therefore, the inverse of the matrix is: ${{A}^{-1}}=\left[ \begin{matrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{6} & 0 \\ 0 & 0 & \frac{1}{9} \\ \end{matrix} \right]$ Where $B={{A}^{-1}}$ Now, check the result for $A{{A}^{-1}}={{I}_{3}}$ And ${{A}^{-1}}A={{I}_{3}}$ Here, $A=\left[ \begin{matrix} 3 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 9 \\ \end{matrix} \right]$ So, \begin{align} & A{{A}^{-1}}=\left[ \begin{matrix} 3 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 9 \\ \end{matrix} \right]\left[ \begin{matrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{6} & 0 \\ 0 & 0 & \frac{1}{9} \\ \end{matrix} \right] \\ & A{{A}^{-1}}=\left[ \begin{matrix} 1+0+0 & 0+0+0 & 0+0+0 \\ 0+0+0 & 0+1+0 & 0+0+0 \\ 0+0+0 & 0+0+0 & 0+0+1 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \\ & ={{I}_{3}} \end{align} And, \begin{align} & {{A}^{-1}}A=\left[ \begin{matrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{6} & 0 \\ 0 & 0 & \frac{1}{9} \\ \end{matrix} \right]\left[ \begin{matrix} 3 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 9 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1+0+0 & 0+0+0 & 0+0+0 \\ 0+0+0 & 0+1+0 & 0+0+0 \\ 0+0+0 & 0+0+0 & 0+0+1 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \\ & ={{I}_{3}} \end{align} Thus, $A{{A}^{-1}}={{I}_{3}}$ and ${{A}^{-1}}A={{I}_{_{3}}}$.
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Compare slopes of functions II
Lesson 13 of 20
Objective: SWBAT solve problems involving linear functions flexibly.
Each day, students complete a warm-up that usually consists of spiraling the previous day's material, in addition to older material. Warm-up problems also sometimes extend lessons that students have encountered before to more unfamiliar contexts.
For a video narrative about how I structure each lesson, and how the warm-up fits in, click here.
Today's warm-up pushes students to move flexibly between multiple representations of a given linear function.
Play of the Day
The homework file is a resource that generally includes 5-7 problems, some of which are related to the day's lesson, as well as spiraled review of previous lessons. I also give the kids the answers to all the problems.
Sometimes, the HW file is a take-home assessment.
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In this chapter, we explore the significance of neighborhood councils as a means of grassroots decision-making and community empowerment. Neighborhood councils are local, participatory bodies that give residents a voice in shaping the policies, projects, and developments that affect their communities. By understanding the importance of neighborhood councils, we can appreciate their role in fostering active citizenship, enhancing community cohesion, and ensuring inclusive decision-making processes.
Local Democracy and Active Citizenship
a. Building Trust and Engagement: Neighborhood councils provide a platform for residents to actively participate in local decision-making processes. By involving community members in shaping their neighborhoods, councils foster trust, engagement, and a sense of ownership among residents.
b. Amplifying Community Voices: Neighborhood councils amplify the voices of marginalized groups and traditionally underrepresented communities. They provide a space for individuals to raise concerns, propose solutions, and advocate for their community's needs and aspirations.
c. Enhancing Democratic Values: Through neighborhood councils, communities learn about democratic principles, such as consensus-building, compromise, and respect for diverse perspectives. This fosters a culture of active citizenship, where residents are encouraged to take responsibility for their neighborhoods.
Inclusive Decision-Making and Community Empowerment
a. Bottom-Up Decision-Making: Neighborhood councils facilitate bottom-up decision-making, ensuring that decisions reflect the genuine needs and aspirations of the community. This approach challenges top-down decision-making processes and empowers residents to shape the future of their neighborhoods.
b. Local Knowledge and Expertise: Community members possess invaluable local knowledge and expertise about their neighborhoods. Neighborhood councils provide a platform for sharing this knowledge and collaborating with local authorities and organizations to make informed decisions.
c. Ownership and Empowerment: Through neighborhood councils, residents feel a sense of ownership and empowerment in improving their communities. They become active agents of change, working collectively to address local challenges, promote initiatives, and enhance the overall well-being of their neighborhoods.
Strengthening Community Cohesion and Social Capital
a. Building Social Connections: Neighborhood councils foster social connections and build social capital within communities. They provide opportunities for residents to interact, collaborate, and form relationships based on shared goals and aspirations.
b. Bridging Divides: By bringing together residents from diverse backgrounds, neighborhood councils bridge social, cultural, and economic divides. They create spaces where individuals can find common ground, build understanding, and work together for the betterment of their neighborhoods.
c. Nurturing Civic Pride: Neighborhood councils instill a sense of civic pride and belonging among residents. By actively participating in decision-making processes and witnessing the positive outcomes of their efforts, residents develop a stronger attachment to their neighborhoods and are more likely to contribute to their ongoing development.
Neighborhood councils play a vital role in grassroots decision-making and community empowerment. They enable residents to actively participate in shaping the policies and projects that impact their neighborhoods. By promoting local democracy, inclusivity, and active citizenship, neighborhood councils foster community cohesion, strengthen social capital, and nurture a sense of ownership and pride among residents. As we embrace the importance of neighborhood councils, we move closer to creating vibrant, resilient, and thriving communities where every voice is heard and valued.
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Immunization Best Way To Protect Infants From Pneumonia
According to new guidelines developed jointly by the Pediatric Infectious Diseases Society and the Infectious Diseases Society of America, the...
According to new guidelines developed jointly by the Pediatric Infectious Diseases Society and the Infectious Diseases Society of America, the best way to prevent infants and young children from developing community-acquired pneumonia (CAP) is to get them vaccinated. These are the first-ever clinical guidelines on the topic and note that tots who are 6 months and older should have annual vaccinations against the influenza virus. The new guidelines note that "Pneumonia is the single greatest cause of death in children worldwide, and that "Each year, >2 million children younger than 5 years die of pneumonia." The recommendations go on to say that the best prevention practices are as follows:
- Children should be immunized with vaccines for bacterial pathogens, including S. pneumoniae, Haemophilus influenzae type b, and pertussis to prevent CAP.
- All infants ≥6 months of age and all children and adolescents should be immunized annually with vaccines for influenza virus to prevent CAP.
- Parents and caretakers of infants <6 months of age, including pregnant adolescents, should be immunized with vaccines for influenza virus and pertussis to protect the infants from exposure.
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WHITE LION SKULL
Details of this fossil’s teeth are shown in the smaller pictures. These 82-million-year-old remains are proofs that evolution never happened. This mammal has exactly the same characteristics today as it had 82 million years ago. Its dimensions, anatomical features and physical properties are identical to those of lions alive today. This poses a major dilemma for the claims made by Darwinists, because far from revealing the transitional forms expected by Darwinists, the fossil record gives up countless species that never changed at all.
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During metaphase, the chromosomes that carry genetic information align in the equator of the cell before they split off into two daughter cells with identical genetic material. Metaphase is the third stage of mitosis, which is a phase of the cell cycle where chromosomes in the nucleus are divided between two cells.Continue Reading
Before metaphase takes place, the protein formations form around the centromere. These protein formations are called kinetochores. Long protein filaments extend from the poles on either end of the cell and attach to the kinetochores. These microtubules pull the sister chromatids back and forth until they are fully aligned down the center of the cell. At this point of the cycle, the cell begins to divide and moves into the fourth stage of the cycle, which is known as anaphase. The metaphase cycle takes up about 4 percent of the cycle of the cell.
During anaphase, the paired centromeres begin to move apart, and once they separate from one another, they are considered a full chromosome, or the daughter chromosomes. These chromosomes move to the poles at the opposite ends of the cell and the kinetochore fibers become shorter. The two cell poles also move further apart in preparation for telophase.Learn more about Cells
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Physics with Calculus/Mechanics/Scalar and Vector Quantities
IntroductionEdit
The study of physics is intimately tied in with the study of mathematics. Sometimes, the direction of a number or quantity is as important as the number itself. Mathematicians in the 19th century developed a convenient way of describing and interacting with quantities with and without direction by dividing them into two types: scalar quantities and vector quantities. Scalar quantities have a magnitude but no direction. Vector quantities have both a magnitude and a direction. For instance, one might describe a plane as flying at 400 miles per hour. However, simply knowing the speed of the airplane is not nearly as useful as knowing the speed and direction of the airplane, so a more accurate description may be a plane flying at 400 miles per hour southeast.
Scalar QuantitiesEdit
Scalar quantities are numbers that have a magnitude but no direction. Scalars are represented by a single letter, such as $a$. Some examples of scalar quantities are numbers with the proper units ((such as three)), mass (five kilograms), and temperature (twenty-two degrees Celsius).
Vector QuantitiesEdit
Vectors are a geometric way of representing quantities that have direction as well as magnitude. An example of a vector is force. If we are to fully describe a force on an object we need to specify not only how much force is applied, but also in which direction. Another example of a vector quantity is velocity -- an object that is traveling at ten meters per second to the east has a different velocity than an object that is traveling ten meters per second to the west. This vector is a special case, however,sometimes people are interested in only the magnitude of the velocity of an object. This quantity, a scalar, is called speed which has magnitude but no given direction.
When vectors are written, they are represented by a single letter in bold type or with an arrow above the letter, such as $\mathbf{A}$ or $\vec A$. Some examples of vectors are displacement (e.g. 120 cm at 30°) and velocity (e.g. 12 meters per second north). The only basic SI unit that is a vector is the meter. All others are scalars. Derived quantities can be vector or scalar, but every vector quantity must involve meters in its definition and unit.
Unit VectorsEdit
An illustration of common choice of unit vectors in a Cartesian coordinate system
Strictly speaking, vectors exist separately from any coordinate systems. As vectors are geometric objects, we do not need to define a coordinate system in order to talk about vectors—or even to perform most operations on vectors. For example, consider the triple of numbers: number of apples, number of bananas, and number of carrots you have. Say that you calculate the triple in one coordinate system and get (1,2,3). If you rotate your coordinate system, and recalculate, you will have (1,2,3) again. Thus, the triple does not have the most important property of a vector -- that is transform like the coordinate system.
Nevertheless, it is often convenient to introduce a coordinate system. In three dimensions, for many problems the rectangular, or Cartesian coordinate system (after French mathematician René Descartes) turns out to be convenient, and this coordinate system can be defined in terms of unit vectors.
A unit vector is a vector pointing in a given direction with a magnitude of one. Essentially, it merely indicates direction. In a Cartesian system the three unit vectors are called i, j, and k (or, in handwriting, with a little "hat" on top, as $\hat{i}$, $\hat{j}$, and $\hat{k}$). Colloquially, you might refer to the directions of the unit vectors as "east", "north", and "up". One could just have easily chosen i as up, j as east, and k as north. In choosing i, j, and k, once i and j are chosen, k must point to a particular direction, so that a common convention called "right-hand rule" holds. Mathematically, this can be compactly expressed as,
$\hat{k} = \hat{i} \times \hat{j}$,
but we will expand more on this as we describe "cross products" later on.
Unit vectors are generally chosen to be orthogonal. That is, each unit vector is perpendicular to each of the others. While unit vectors do not need to be orthogonal, working with a coordinate system defined by orthogonal unit vectors will be convenient in most cases. There are two other major coordinate systems used in physics—cylindrical coordinates and spherical coordinates. These will be introduced at a later time as necessary. unit vector has a magnitude of one and it has infinetely meaning in xand y
Vector ComponentsEdit
Every vector may be expressed as the sum of its n unit vectors.
$\vec{A} = a_x~\hat{i} + a_y~\hat{j} + a_z~\hat{k}$
The quantities ax, ay, and az are called the vector components of vector A. Sometimes they are represented simply as an ordered triple (e.g. (ax,ay,az)) especially when the choice and ordering of three unit vectors are not ambiguous.
Vector AlgebraEdit
NegationEdit
Illustration of vector negation and scalar multiplication
$-\vec{A} = -(a_x~\hat{i} + a_y~\hat{j} + a_z~\hat{k}) = -a_x~\hat{i} - a_y~\hat{j} - a_z~\hat{k}$
Considering a vector represented graphically by an arrow, the negative of a vector would be represented by a vector of the same length but opposite direction.
Scalar MultiplicationEdit
$k\vec{A} = ka_x~\hat{i} + ka_y~\hat{j} + ka_z~\hat{k}$
Note that vector negation is merely multiplication by a scalar, where that scalar is -1. A scaled vector represented graphically would point in the same direction as the original vector but have its magnitude scaled by a factor of k.
\begin{align}\vec{A} + \vec{B} &= (a_x~\hat{i} + a_y~\hat{j} + a_z~\hat{k}) + (b_x~\hat{i} + b_y~\hat{j} + b_z~\hat{k}) \\ &= (a_x + b_x)~\hat{i} + (a_y + b_y)~\hat{j} + (a_z + b_z)~\hat{k}\end{align}
Two vectors can be added graphically by placing the tail of the second vector (here, B) coincidental with the tip of the first vector (A). The resultant vector A + B is the vector drawn from the tail of A to the tip of B.
Any number of vectors can be added in this fashion. Vector addition is commutative:
$\vec{A} + \vec{B} + \vec{C} = \vec{C} + \vec{B} + \vec{A}$
and associative:
$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$
Dot ProductEdit
When we multiply two vectors, we can either apply a multiplication rule that produces a scalar as the end result, or one that produces a vector as the end result. The first one that produces a scalar is called dot product. In mathematical texts, this is often called inner product, and some older texts will refer to this as scalar product (not to be confused with scalar multiplication); they are all the same. Dot product has all the usual properties of products, such as associativity, commutativity, and the distributive property. Geometrically, dot product is defined as:yirga egzihaye
$\vec{A}\cdot \vec{B} = A B \cos(\theta)$,
where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$. Note that since $\cos(0) = 1$, if $\vec{A}$ is parallel to $\vec{B}$, then $\vec{A} \cdot \vec{B} = AB$. On the other hand, since $\cos(90^\circ) = 0$ if $\vec{A}$ is perpendicular to $\vec{B}$, then $\vec{A} \cdot \vec{B} = 0$. Using this as the guiding rule, we find below relationship:
\begin{align} \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1 \\ \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0\end{align}.
Using this, we can define dot product in terms of component vectors as follows:
$\vec{A}\cdot\vec{B} = (A_x~\hat{i} + A_y~\hat{j} + A_z~\hat{k})\cdot(B_x~\hat{i} + B_y~\hat{j} + B_z~\hat{k}) = A_x B_x + A_y B_y + A_z B_z$.
You are encouraged to expand out the multiplication explicitly, using the distributive property and find which terms cancel to zero and which products become 1.
Cross ProductEdit
The second multiplication rule for product of two vectors yields yet another vector. This multiplication rule is a very special one—in fact, it is a special property of 3-dimensional space that we can define a vector multiplication is this way and still obtain a vector. This rule will not work when limited to 2-D, and in any dimensions greater than 3, an extension of this rule will not result in another vector (cf. dot product can be naturally extended or limited to any dimensions to produce a scalar). This multiplication is called cross product, and in other texts, you may find terms outer product and vector product. The product can be defined with the two rules, first specifying the product vector's direction, and the second specifying its magnitude:
1. $\vec{A}\times\vec{B}$ is perpendicular to $\vec{A}$ and $\vec{B}$ (that is, perpendicular to the plane defined by these two vectors). This leaves two possible directions along the line perpendicular to the plane. One of the two directions is called by a "right-hand rule": Hold out index finger, middle finger, and the thumb so that they are all perpendicular to each other. Let the index finger point towards direction of $\vec{A}$, and the middle finger towards $\vec{B}$. Then the thumb points towards the direction of $\vec{A}\times\vec{B}$. The ordering is important here (note exchanging A and B makes the thumb point in the opposite direction).
2. $|\vec{A} \times \vec{B} | = A B \sin(\theta)$, where $\theta$ is again the angle between $\vec{A}$ and $\vec{B}$.
Applying this definition to unit vectors again, we find following relationships:
\begin{align} \hat{i} \times \hat{j} &= - \hat{j} \times \hat{i} = \hat{k} \\ \hat{j} \times \hat{k} &= - \hat{k} \times \hat{j} = \hat{i} \\ \hat{k} \times \hat{i} &= - \hat{i} \times \hat{k} = \hat{j} \\ \hat{i} \times \hat{i} &= \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0 \end{align}.
And in terms of components, we have (after a tedious algebra):
$\vec{A} \times \vec{B} = (A_y B_z - B_y A_z)~\hat{i} + (A_z B_x - B_z A_x)~\hat{j} + (A_x B_y - B_x A_y)~\hat{k}$.
It turns out we can write this complicated relationship as a determinant of a 3 x 3 matrix:
$\vec{A} \times \vec{B} = \left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{matrix} \right|$.
Some properties of cross product, such as $\vec{A} \times \vec{B} = - \vec{B} \times \vec{A}$ and $\vec{A} \times \vec{A} = 0$ can be derived as a property of the determinant of the matrix.
Useful Properties of Dot Product and Cross ProductEdit
Both the dot product and the cross product distribute over vector addition.
$\vec{A}\cdot(\vec{B}+\vec{C})=\vec{A}\cdot\vec{B}+\vec{A}\cdot\vec{C}$
$\vec{A}\times(\vec{B}+\vec{C})=\vec{A}\times\vec{B}+\vec{A}\times\vec{C}$
$(\vec{A}+\vec{B})\times\vec{C}=\vec{A}\times\vec{C}+\vec{B}\times\vec{C}$
The dot product of two vectors is proportional to the cosine of the angle between them, and their cross product is proportional to the sine of the angle between them.
$\vec{A}\cdot\vec{B}=\|\vec{A}\|\|\vec{B}\|\cos(\theta)$
$\|\vec{A}\times\vec{B}\|=\|\vec{A}\|\|\vec{B}\|\sin(\theta)$
As we have seen already, the dot product is associative and commutative.
$\vec{A}\cdot(\vec{B}\cdot\vec{C})=(\vec{A}\cdot\vec{B})\cdot\vec{C}$
$\vec{A}\cdot\vec{B}=\vec{B}\cdot\vec{A}$
It is important to remember that the cross product has neither of these properties. Instead of being commutative, it is anticommutative.
$\vec{A}\times\vec{B}=-\vec{B}\times\vec{A}$
The cross product is not even associative. For example, consider $(\vec{A}\times\vec{A})\times\vec{B}$. Since the sine of the angle between $\vec{A}$ and itself is 0, $\vec{A}\times\vec{A}=\vec{0}$, and so $(\vec{A}\times\vec{A})\times\vec{B}=\vec{0}$. On the other hand, $\vec{A}\times(\vec{A}\times\vec{B})$ is not zero, since $\vec{A}$ and $\vec{A}\times\vec{B}$ are perpendicular. In fact, if $\vec{A}$ and $\vec{B}$ were perpendicular, its direction would be opposite to that of $\vec{B}$. Check this yourself using the right hand rule.
The component of $\vec{A}$ parallel to $\vec{B}$ is given by
$\left(\frac{\vec{B}\cdot\vec{A}}{\vec{B}\cdot\vec{B}}\right)\vec{B}$
and the perpendicular component of $\vec{A}$ is given by
$\left(\frac{\vec{B}\times\vec{A}}{\vec{B}\cdot\vec{B}}\right)\times\vec{B}$.
This leads to some interesting properties involving combinations of the products, such as
$(\vec{B}\cdot\vec{B})\vec{A}=(\vec{A}\cdot\vec{B})\vec{B}+(\vec{B}\times\vec{A})\times\vec{B}$,
$\vec{A}\times(\vec{B}\times\vec{C})=(\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C}$, and
$(\vec{A}\times\vec{B})\cdot(\vec{C}\times\vec{D})=(\vec{A}\cdot\vec{C})(\vec{B}\cdot\vec{D})-(\vec{A}\cdot\vec{D})(\vec{B}\cdot\vec{C})$.
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The fundamental British values, first set out by the government in the ‘Prevent’ strategy in 2011 and reinforced through further Department for Education advice in November 2014 are:
- The Rule of Law
- Individual Liberty
- Mutual respect and tolerance of those different faiths and beliefs
Actively promoting the values means challenging opinions or behaviours in the PRU that are contrary to fundamental British values. The Teachers Standards expect teachers to uphold public trust in the profession and maintain high standards of ethics and behaviour, within and outside PRU. This includes not undermining fundamental British values.
The following list describes the understanding and knowledge expected of pupils as a result of our PRU promoting fundamental British values:
- An understanding of how citizens can influence decision making through democratic processes.
- An understanding that there is a separation of power between the executive and the judiciary, and that while some public bodies such as the police and the army can be held to account through Parliament, others such as courts maintain independence.
- An appreciation that living under the rule of law protects individual citizens and is essential for their wellbeing and safety.
- An understanding that the freedom to hold other faiths and beliefs is protected in law.
- An acceptance that people having different faiths or beliefs to oneself (or having none) should be accepted and tolerated, and should not be the cause of prejudicial or discriminatory behaviour.
- An understanding of the importance of identifying and combatting discrimination.
Below are some examples of actions we take to promote British values at SSPS:
- Curriculum -Across the curriculum, opportunities are built into lessons for students to explore and practice fundamental British values, either through topics studied or by following the PRU’s general structures and behaviour codes.
- Wider PRU Life – Our PRU vision, values and general ethos support fundamental British values and we have a strong emphasis on the development of SMSC across the PRU. Within the PRU, pupils are actively encouraged to make choices, knowing that they are in a safe and supportive environment.
- Student Voice – We promote democratic processes, fostering the concept and application of freedom of speech and group action to address needs and concerns
- Ongoing Opportunities – We use local and national opportunities that arise to promote fundamental British values and provide pupils with the opportunity to learn how to argue and defend points of view.
- Extra – Curricular – There is a wide range of sporting, creative and academic activities to choose from including outdoor education, Duke of Edinburgh Award, art, cookery, motor vehicle engineering and construction. These promote self – development, self – esteem, confidence and understanding of the concept of fair play, following and developing rules, inclusion, celebrating and rewarding success.
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## Thursday, September 4, 2008
### Voltage and current in a practical circuit BASIC CONCEPTS OF ELECTRICITY
Because it takes energy to force electrons to flow against the opposition of a resistance, there will be voltage manifested (or "dropped") between any points in a circuit with resistance between them. It is important to note that although the amount of current (the quantity of electrons moving past a given point every second) is uniform in a simple circuit, the amount of voltage (potential energy per unit charge) between different sets of points in a single circuit may vary considerably:
Take this circuit as an example. If we label four points in this circuit with the numbers 1, 2, 3, and 4, we will find that the amount of current conducted through the wire between points 1 and 2 is exactly the same as the amount of current conducted through the lamp (between points 2 and 3). This same quantity of current passes through the wire between points 3 and 4, and through the battery (between points 1 and 4).
However, we will find the voltage appearing between any two of these points to be directly proportional to the resistance within the conductive path between those two points, given that the amount of current along any part of the circuit's path is the same (which, for this simple circuit, it is). In a normal lamp circuit, the resistance of a lamp will be much greater than the resistance of the connecting wires, so we should expect to see a substantial amount of voltage between points 2 and 3, with very little between points 1 and 2, or between 3 and 4. The voltage between points 1 and 4, of course, will be the full amount of "force" offered by the battery, which will be only slightly greater than the voltage across the lamp (between points 2 and 3).
This, again, is analogous to the water reservoir system:
Between points 2 and 3, where the falling water is releasing energy at the water-wheel, there is a difference of pressure between the two points, reflecting the opposition to the flow of water through the water-wheel. From point 1 to point 2, or from point 3 to point 4, where water is flowing freely through reservoirs with little opposition, there is little or no difference of pressure (no potential energy). However, the rate of water flow in this continuous system is the same everywhere (assuming the water levels in both pond and reservoir are unchanging): through the pump, through the water-wheel, and through all the pipes. So it is with simple electric circuits: the rate of electron flow is the same at every point in the circuit, although voltages may differ between different sets of points.
### Related Posts by Categories
Electronics Hoctro | Electrical
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Since it’s hard to get away from it in the news just now, many might be wondering about the origins of impeachment. It’s an interesting story, and it shows up something important about the distinctive constitutional development of the United States.
The origins of impeachment go back to the Middle Ages in England. But after a long period of disuse, it was revived in the early seventeenth century as a way for parliament to hold the servants of King James I accountable for corruption and other offences. First minor officials, then, as the opposition became bolder, more senior ones, were accused by the House of Commons and tried by the House of Lords.
Then, in 1628, when James’s son Charles I was king, the Commons turned to impeachment to try to remove his chief minister, the Duke of Buckingham – not because he had committed crimes, but because he was both incompetent and pursuing policies that they disagreed with.
The Lords, however, were unwilling to convict someone without proof of an actual crime. Buckingham was soon assassinated, but the problem returned in 1641, when the opposition was set on the removal of another chief minister, the Earl of Strafford. They impeached him for high treason, arguing that by aiding the king’s autocratic policies he had attempted to subvert the fundamental laws of the kingdom.
Again, securing a conviction in the Lords proved difficult. Parliament circumvented the problem by passing an act simply declaring Strafford to be guilty, and he was duly beheaded – but this was clearly a remedy appropriate for only the most extreme circumstances. It’s perhaps not surprising that he following year the country descended into civil war.
What parliament wanted was the ability to control the policy direction of the king’s government, and that meant the ability to change its personnel on policy rather than legal grounds. As the House of Commons explained in the Grand Remonstrance:
It may often fall out that the Commons may have just cause to take exceptions at some men for being councillors, and yet not charge those men with crimes, for there be grounds of diffidence which lie not in proof. There are others, which though they may be proved, yet are not legally criminal.
The king and his supporters, of course, were having none of this. And after an interval of civil war, republic and restoration, the problem recurred. In 1678 another chief minister, the Earl of Danby, was impeached for treason; he kept his head, but he spent five years in prison while parliament and Charles II wrangled over his fate.
The “Glorious Revolution” of 1688 improved parliament’s position, but it didn’t remove disagreement between monarch and parliament. And impeachment remained at best a clumsy mechanism for resolving it. Few impeachments ever even proceeded to trial; they were used, by both sides, to force political opponents from office or to discredit them so as to prevent their return.
Things started to change after Queen Anne came to the throne in 1702. Embroiled in a major war with France she was constantly in need of parliamentary support, and she was less able than her predecessors to withstand pressure from the (male, of course) party leaders. Gradually they secured the upper hand, forcing the queen to choose ministers from the party that could deliver a parliamentary majority, regardless of her personal preferences.
Party government made impeachment unnecessary. Ministers who had the support of a majority party were safe from it; those who did not could be removed by a simple vote in parliament (what we now call a vote of no confidence), because otherwise government could not be carried on. And once in office, a party had considerable power to consolidate a parliamentary majority by distribution of government jobs and other favors.
The last “political” impeachments came in 1715, when the Tory ministers who had lost office on Anne’s death the previous year were accused of treason. But after two years their Whig opponents, now securely in power and fighting among themselves, gave up the attempt to prove the charges. From then on, impeachment was confined to cases of actual non-political crimes; the last occasion was in 1806.
So by the time that the United States constitution was written in 1787, it was well-established British doctrine that impeachment was about serious offences by public officials, not just political disagreement. But since theory tends to lag behind practice, the underlying reasons were still poorly understood.
As a result, the Americans created a system based on British practice of a century or so earlier, which politicians of the time continued to pay lip service to. In addition, they adopted (article I, section 6.2) a measure that backbench politicians in Britain had always been keen on but had never been able to implement: the exclusion of office-holders from parliament.
The intent of that was to fight corruption – to stop the government from being able to build up a compliant majority by distributing jobs. But it also prevented parliamentary government of the sort that had developed (and was still developing) in Britain. Other ways would have to be found for resolving disagreements between the head of state and the legislature.
How that worked out, and what role impeachment played in it, will be the topic of part II.
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### OHM’s LAW EXERCISE2
Exercise2
1. Suppose that the dc generator in Figure 1 produces 10 V and the potentiometer is set to a value of 10 ohms. What is the current?
2. Imagine that dc generator in Figure 1 produces 100 V and the potentiometer is set to 10 kiloohms. What is the current?
3. Suppose that dc generator in Figure 1 is set to provide 88.5 V, and the potentiometer is set to 477 Megaohms. What is the current?
4. Suppose the potentiometer in Figure 1 is set to 100 ohms, and the measured current is 10 mA. What is the dc voltage?
5. Adjust the potentiometer in Figure 1 to a value of 157 kiloohms, and suppose the current reading is 17.0 mA. What is the voltage of the source?
6. Suppose you set the potentiometer in Figure 1 so that the meter reads 1.445 A, and you observe that the potentiometer scale shows 99 ohms. What is the voltage?
7. If the voltmeter in Figure 1 reads 24 V and the ammeter shows 3.0 A, what is the resistance of the potentiometer?
8. What is the value of the resistance in Figure 1 if the current is 18 mA and the voltage is 229 mV?
9. Suppose the ammeter in Figure 1 reads 52 uA and the voltmeter indicates 2.33 kV. What is the resistance?
10. Suppose that the voltmeter in Figure 1 reads 12 V and the ammeter shows 50 mA. What is the power dissipated by the potentiometer?
11. If the resistance in the circuit of Figure 1 is 999 ohms and the voltage source delivers 3 V, what is the power dissipated by the potentiometer?
12. Suppose the resistance in Figure 1 is 47 kiloohms and the current is 680 mA. What is the power dissipated by the potentiometer?
13. How much voltage would be necessary to drive 680 mA through a resistance of 47 kiloohms?
14. Consider five resistors in parallel. Call them R1 through R5. Let the resistance values be as follows: R1 = 100 ohms, R2 = 200 ohms, R3 = 300 ohms, R4 = 400 ohms, and R5 = 500 ohms. What is the total resistance?
15. cont.... What is the total conductance?
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Music of the Holocaust
Some of the most recognizable classics in music literature are of German origin. Handel's Messiah, Bach's Magnificat, Beethoven's Ninth Symphony, Brahm's Lullaby, Wagner's Ring series, Strauss's Der Rosenkavalier, Robert Schumann's Fantasia, and Schubert's Unfinished Symphony are but a few.
This rich musical heritage was used by Hitler to promote Aryan superiority. His ideas concerning music and art shaped the cultural atmosphere and political policies for all of Germany. All compositions written by Jews or by those persons suspected of being sympathizers were banned. It became unlawful for artists and musicians to perform in public without being first a member of the state sanctioned Reichsmusikkammer (Reich Music Chamber or RMK). Anyone who defied the law was arrested.
Many artists and musicians were government employees, hired in various capacities to create and disseminate Aryan culture. By 1939, the RMK leaders spoke of the elimination of the Jews from the cultural life of the people; exceptions were made for performances by prominent Jews from other countries. Jazz music was banned, as it was considered to be "non-Aryan Negroid." Control and censorship of all radio broadcasts were implemented, with only approved nationalistic music allowed. All other music was prohibited and labeled "entarte" or degenerate.
- Music of the Ghettos and Camps. Although the inhabitants were incarcerated, music was composed and performed giving voice to the indomitable human spirit within the ghettos and camps. Most cruelly, the large camps had orchestras and bands who were forced to play while their families, friends and neighbors were selected for death then sent to the gas chambers or firing squads.
- Music of the Third Reich. Hitler was decidedly against any music that did not have the Teutonic overtones of Wagner and Bruckner. State sanctioned music had to sound German.
- "Degenerate" Music. Like their counterparts in the Arts, musicians were trying to express through music the world around them. Any music that demonstrated abstract expressionism, jazz, or experimented with "atonality" was prohibited and labeled "entarte" or degenerate.
- Music in Response to the Holocaust. Composers and musicians have created a vast array of music in response to the Holocaust. Written and performed both during and after the Holocaust, the requiems, operas, cantatas and ballads are filled with the stories of the victims, survivors and resistance fighters. Some are poems found in ghettos and camps that have been set to music while others are in remembrance of specific victims like Anne Frank.
- Teacher Resources. Here you will find lesson plans and other resources for the study of the Holocaust through music.
| Ghettos & Camps | Reich Music | "Degenerate" Music | Response | Teacher Resources |
A Teacher's Guide to the Holocaust
Produced by the Florida Center for Instructional Technology,
College of Education, University of South Florida © 1997-2013.
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We have developed multiple design focused curricula for our projects. Our curricula provide students with opportunities to explore science ideas within a meaningful context to solve a design challenge integrating science content and practices. Students work like scientists and engineers to engage in research, collect and interpret data, and construct scientific arguments and explanations based on evidence.
GROWING HEALTHY PLANTS!
Growing enough plants to feed an increasing global population can be very difficult. Besides providing the right soil, nutrients, and water, farmers must deal with a variety of issues, such as invasive species, disease-causing bacteria and fungus, pests, and climate change. Additionally, there is limited fertile land to grow food crops. Thus, growing lots of food for an increasing population requires a lot of science and engineering! The design challenge for the Growing Healthy Plants unit asks students to make recommendations for growing healthy plants, such as the traits, or characteristics, that healthy plants should have and the kinds of environments that can help them grow. To solve their challenge, students participate in hands-on activities and virtual experiments to learn about growing healthy plants. They explore how environmental factors, such as soil, light, and water, as well as plants’ genetics influence plants’ growth and reproduction to create a habitat to grow as much food as possible using limited space and resources to feed a growing population. Along with engaging in a wide range of science practices, students learn about processes of growth and development of organisms, variation and inheritance of genetic traits, and the interactions between living and non-living things in ecosystems.
MAKE YOUR OWN COMPOST!
Americans generate a lot of trash each year, approximately 250 million tons! A large percentage of this trash ends up in landfills. Aside from the negative impact landfills have on the environment, such as emitting nearly 6.3 million metric tons of methane into the atmosphere, it is also very expensive to move trash around and safely manage such a mammoth waste stream. However, many things we throw into landfills can be used to make compost, which can provide valuable nutrients to our gardens and farms. While scientists and engineers know a lot about the composting process, there are still many unanswered questions about how to engineer more effective composting systems. To help minimize the amount of waste going into landfills students are challenged to work together to create composting program for their school that ensures that the compost breaks down quickly and contains a lot of nutrients. To solve this challenge, students move back and forth between conducting physical and virtual experiments, using a physical soda bottle bioreactor and a compost simulation, to learn about the science of composting. Through conducting these investigations, students learn about the factors that affect decomposition along with learning about the role of organisms in ecosystems and the cycling of matter and the transformation of energy in ecosystems.
INCLINED PLANE DESIGN CHALLENGE!
Understanding the physics of inclined planes can help us to decrease the amount of force needed to do work and more easily accomplish a multitude of real-world tasks, such as designing ramps for people with limited mobility to gain access to buildings. The premise for the inclined plane design challenge is that students are borrowing a mini pool table from a friend to use at a birthday party. They need to figure out how to get the mini pool table into a moving van to get it to their house using the least applied force and work. To solve the challenge, students set up and conduct several physical and virtual inclined plane experiments to learn about how factors such as the height and length of an inclined plane, along with the amount of friction on its surface, affects the amount of applied force and work it takes to move the mini pool table. They also explore the relationships between energy and work, mechanical advantage and applied force, and the trade-off between force and distance when doing work with any simple machine.
PULLEY DESIGN CHALLENGE!
Whether we think about it or not, we rely on pulleys every day to get work done! From the cranes towering over buildings at construction site to the lifting and lowering our window blinds, pulleys help us to do tasks that would be very difficult to accomplish without them. In our pulley unit, students are challenged to design a pulley system to lift a very heavy mascot statue on to a pedestal located on the front lawn of the school using the least amount of applied force possible. Students work toward solving this challenge by setting up and conducting several physical and virtual pulley experiments to test their ideas for the best design to lift the statue and meet the design constraints. For example, they test several independent variables, such as the type of pulley system and the height to lift the mascot statue, to understand how the changes in the independent variables affects the applied force, potential energy, work, and mechanical advantage. Exploring these relationships helps them to learn about the force – distance trade-off when using simple machines to reduce the amount of force needed to accomplish difficult tasks.
ROLLER COASTER DESIGN CHALLENGE!
Many people love the rush of adrenaline that they experience on a roller coaster ride! But few people think about the physics involved in designing roller coasters that optimize fun while ensuring riders’ safety. The roller coaster design challenge is focused on helping the Gonzales family, who own and run a large amusement park. In recent years, attendance at the Gonzales’ amusement park has been dropping and they want to add a new roller coaster to increase park attendance. Students are challenged to act as scientists and engineers to design an exciting but safe roller coaster and to submit proposals to the Gonzales family providing an explanation about their roller coaster design, the physics behind their choices, and why their roller coaster is both fun and safe. To learn about the physics underlying the fun and safely of roller coasters, students perform multiple experiments using a roller coaster simulation to test their ideas. They learn about physics concepts like force, motion, and energy. They also explore the relationships between several factors, such as the height and shape of the different parts of the roller coaster and how changing these this affects the amount of velocity and acceleration riders’ experience.
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DISASTER WORK AND STRESS
George W. Doherty, M.S.
Stress is a natural phenomenon. It is not necessarily negative. The body needs certain levels of stimulation and stress in order to be able to function. Stress is the body's normal, adaptive response to the environment. A stressor is the actual event which produces a demand or stress on an individual. The resulting wear and tear on an individual is called strain (Mitchell & Resnik, 1981).
Stress and distress are two different things. When an individual is distressed it is due to a disorder in their adaptation to stressors. Stressors fall into a number of categories. Charlesworth and Nathan (1984) have identified the following, some of which are positive stressors:Cognitive
The following, generally considered to be positive events, are also considered stressors:Change
Increased energy and endurance
The stress reaction prepares the body for change. It puts the body on alert to respond and adapt. This reaction goes back to a much earlier time in the evolution of the human species. The need to adapt to the environment still exists, but the physical dangers no longer exist as they once did. However, the stress reaction provokes the same biochemical reactions.
The part of the brain known as the hypothalamus alerts the nervous system to release energy when faced with physical danger or threat. The nervous system alerts the endocrine system to send large amounts of hormones into the blood stream, mobilizing the body for action. Muscles tighten up, blood sugar rises, adrenalin and noradrenalin provide emergency energy. This all prepares the body for what Cannon (1929) termed the "fight or flight response".
Hans Selye (1956, 1978) labeled three phases of this normal defense reaction of the body "The General Adaptation Syndrome". General because the consequences of the stressors have effects on several areas of the body. Adaptation refers to its stimulation of defenses designed to help the body adjust or deal with the stressors. Syndrome indicates that the individual pieces of the reaction occur more or less together and are at least partially interdependent. These reactions are similar for all forms of animal life. The physiological responses are the same whether the stressor produces fear, anger, or anxiety (Mitchell & Resnik, 1981).
While the physiological effects occur whether the stress is positive or negative, the psychological effects depend on the type of stress. Excitement, joy and high self-esteem are associated with positive stress. This extra charge of energy, for a short time, produces a controlled form of intense concentration called "eustress". Some examples of eustress include athletes striving to win, surgeons operating for long hours, and marathons. Such examples of eustress contribute to individual excellence. The benefits of optimal stress, if handled properly, include: opportunity for increased growth and maturity, independence, and control.
ONGOING OR LONG-TERM STRESS EFFECTS
Following long continued exposure to the same stressor(s), to which the body has become adjusted, adaptation energy is finally exhausted. This results in what Selye called "diseases of adaptation". These include asthma, chest and back pains, migraines, neuroses, psychoses, skin rash, and others (Selye, 1956; Cox, 1978). Selye pointed out that if stressors did not diminish over a certain period of time, the organism would move from a state of alarm into a state of exhaustion. With continued exhaustion, severe illness may occur. Finally, if signs of the alarm reaction reappear, resistance is gone. The situation becomes irreversible, and the individual dies.
The following are some of the health effects that continued strain, wear and tear can have on individuals (Mitchell & Resnik, 1981; Davis et al, 1982; Charlesworth & Nathan, 1984):Decrease in the effectiveness of the body's immune system, with an increase in colds, flu, and other communicable diseases.
high blood pressure
Gastrointestinal upsets, diarrhea, ulcers, colitis
Muscle tension, strains, backaches, and back injuries
Increased problems with allergies, skin conditions, asthma
Possibly increased vulnerability to heart disease, diabetes, cancer
Weight loss or gain
Increase in use of alcohol, tobacco and other drugs
Psychological difficulties: depression, withdrawal, apathy; or anger, irritability, huperexcitability
Most of the time the stress reactions elicited are so mild that they go unnoticed. Everyone has experienced events which were intensely stressful for a brief period of time. However, once the threat has passed, systems return to normal. Such an isolated stress event, in spite of the internal havoc it raised, probably resulted in absolutely no long-lasting physical damage to the body. On the other hand, a constant state of agitation can result in serious health dysfunctions (Ivancevich & Matteson, 1980). While mild stress and short-term infrequent intense stress produce no lasting harm, constant stress, or acute stress, results in a step-by- step exhaustion of the body's fuel reserves, with the end result - burnout.
Burnout is an effect of long-term stress. It shows up most commonly in the level of an individual's work performance. Farberow & Gordon (NIMH, 1978) have defined burnout as a state of exhaustion, irritability, and fatigue which markedly decreases an individual's effectiveness and capability.
Burnout is an advanced stage of stress. It occurs when there is chronic stress over a long period of time. Burnout can be defined as "collapse of the human spirit". Emotional exhaustion is another way to define it. Some of the early signs of burnout are the same as those for advanced stages of stress: fatigue, sleep disturbances, negative attitude, disillusionment, lowered resistance to infection, hypertension, headache, and stomach disturbances.
IMPACT OF STRESS
The impact of stress depends on a number of factors. Three of these are one's general health, genetics, and prior exposure to stressors. These factors may strengthen and support a disaster worker, resulting in mitigation or softening of the emotional consequences of a disaster. On the other hand, they may place the worker at risk for stress reactions.
The relationship of stress to physical and psychological health has been documented extensively in literature and research. Selye's (1956) research has demonstrated that stressors can cause changes in the immune system, thus wearing down resistance. A super-abundance of hormones secreted can considerably reduce immunity to infection.
Evidence suggests that individuals with certain physical or psychological characteristics are more at risk as potential victims of stress disorders. For example, Type A individuals constantly strive to attain achievements. They are competitive and hard driving. They strive to accomplish more and more in less time. They are chronically impatient with people and situations which they perceive as thwarting their attempts.
Type B individuals, on the other hand, are characterized by the absence of these behaviors. They are relatively relaxed and easygoing, even though they too may be goal-oriented. They appear to have a protective shield which allows them to experience less stress.
EXPOSURE TO PREVIOUS STRESSORS
It is a recognized fact that illness is due to external viruses and pathogenic agents entering the body. However, there is increasing evidence that illness is also due to the eventual broken-down state of the body. After long and continuous exposure to stress (or intermittent periods of intensive stress) hypertension, coronary heart disease, diabetes and ulcers occur (Cox, 1978; Beehr & Newman, 1978).
Research has demonstrated that stressful experiences can make animals more or less vulnerable to a number of cancer tumors, and researchers can speed up the time at which the tumors appear by controlling the number of times the animals are exposed to stress. The stressors applied in these studies resemble many human experiences of stress, such as forced restraint, crowding, handling, shock and noise (as opposed to a non-demanding more protected environment). Over time, the body parts break down.
On the other hand, research has demonstrated that less severe physiological damage to the body occurs when the following circumstances are present: fewer major life changes, socially supportive relationships, experience in handling stress, immunity as a result of many experiences with stress, and high self-esteem (Klein, 1971; House, 1980).
ADDITIONAL FACTORS AFFECTING STRESS AMONG DISASTER WORKERS
Four groups of factors can affect the stress levels experienced by disaster workers. These include:A. Individual FactorsHealth
Previous traumatic experiences
Prior disaster experience
Identity and self-expectations
Perception and interpretation of the event
B. Interpersonal FactorsStrength of social support system
Pre-existing stresses in relationships
Expectations and needs of others
States of family members in disaster
C. Community FactorsSize of community
Previous degree of social solidarity
Prior disaster experiences
Amount of social disruption due to disaster
D. General Aspects of the Disaster or Type of EventWarning
Contrast of scene
Type of disaster
Nature of the destructive agent
Degree of uncertainty
Time of occurrence
Duration of disaster or continued threat
Scope of disaster
Location of disaster
There are three major sources of stress which disaster workers face in their work. These are: Personal loss or injury; Traumatic stimuli; and Mission failure or human error. Each of these can contribute significantly to the stress reactions workers experience during or after a disaster event.
Occupational stressors also affect workers in disasters. Disaster work involves some pretty heavy professional responsibilities. The stakes are high and often involve life or death. Public as well as self-expectations for workers are high. Emergency responses are immediate, continuous and often without letup. There are significant physical, mental and emotional demands placed upon workers under extremely adverse chaotic and traumatic conditions. The physical properties of the work environment can cause additional stress. These include: work area, amount of contact with victims (injured, dead and dying), weather, hazards, work conditions, living conditions, human resources, frustrations and bystanders.
There are also organizational stressors that can occur due to the nature of the emergency organization. Among these are stresses due to: differences among professional vs volunteer organizations; day-to-day vs disaster responsibilities (Warheit, 1970), role clarity and role conflict (Garaventa, 1984); the size of the organization (Garaventa, 1984); rank of the individual in the organization (Kahn et al., 1964; Schein, 1965); chain of command; organizational conflict; and rewards.
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# What is the domain and range of abs(cos x)?
Jul 4, 2016
Domain of $f = \mathbb{R} .$
Range of $f$ = $\left[0 , 1\right] .$
#### Explanation:
Let $f \left(x\right) = | \cos x |$
Since $x$ can take any value, Domain of $f = \mathbb{R} .$
Now we know that, $\forall x \in \mathbb{R} , - 1 \le \cos x \le + 1.$
Therefore,$0 \le | \cos x | \le 1.$ Hence, Range of $f$ =[0,1]#
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- On this page:
Figure 1: Photo Credit - Ginger Holser
The raccoon (Procyon lotor) is a native mammal of Maine, measuring about three feet long, including its 12-inch, bushy, ringed tail. Because its hind legs are longer than the front legs, the raccoon has a hunched appearance when it walks or runs. Each of its front feet has five dexterous toes, allowing raccoons to grasp and manipulate food and other items. (Fig. 1)
Raccoons prefer forested areas near a stream or water source, but have adapted to various environments throughout the state. Raccoon populations can get quite large in urban areas, owing to restrictions on and trapping, lack of predators, and food supplied by humans.
Adult raccoons weigh 15 to 40 pounds, their weight being a result of genetics, age, available food and habitat location. Males have weighed in at over 60 pounds. A raccoon in the wild will probably weigh less than the urbanized raccoon that has learned to live on handouts, pet food and garbage-can leftovers.
As long as raccoons are kept out of human homes, are not cornered, and are treated as wild animals rather than pets, they are not dangerous.
Because raccoons manipulate and moisten food items in water, there is a misconception that raccoons "wash" their food before eating it. However, when water is not available, raccoons use many of the same motions in handling food. (Fig. 1)
Facts about Raccoons
Food and Feeding Behavior
Den Sites and Resting Sites
Reproduction and Home Range
Mortality and Longevity
Raccoons are usually active at night, but can occasionally be spotted during the day eating, searching for food, or napping in a tree. Coastal raccoons take advantage of low tides, whether during the day or night, to forage for shellfish and other food. Once nightly temperatures fall below 25 degrees F, raccoons retreat to their dens, but may occasionally be seen during warm spells in late fall and early spring.
Raccoons take advantage of trails that other wildlife or humans have made, particularly those next to water or in the shelter of woodlands or overgrown fields. They also use culverts to move safely from one side of a road to the other. With a marsh on one side and woods on the other, the culvert becomes the chief route back and forth. In developed areas, raccoon travel along fences, next to buildings, and near food sources.
Figure 2: Drawing Credit - Pandell and Stall
Tracks, Scratch Marks, and Similar Signs
Look for tracks in sand, mud, or soft soil, particularly at either end of a culvert. Also check deck railings, fire escapes, and other surfaces that raccoons use to gain access to structures. (Fig. 2) Tracks may appear as smudge marks on the side of a house where a raccoon shimmies up and descends a downspout or utility pipe.
Sharp, non-retractable claws and long digits make raccoons good climbers. Like squirrels, raccoons can rotate their hind feet 180 degrees and descend trees headfirst. (Cats have claws that do not rotate, they have to back down trees) Scan for scratch marks on trees and other structures that raccoons climb.
Look for wear marks, body oil, and hairs on wood and other rough surfaces, particularly around the edges of den entrances. The den's entrance hole is usually at least four inches high and six inches wide.
Both front and back feet have five toes. The rear foot, which shows the "heel," looks like a small human footprint; the hind tracks are three to four inches long. The front prints have shorter heel marks and are two to three inches long. (Fig. 2)
Note: Raccoon droppings may carry a parasite that can be fatal to humans. Do not handle or smell raccoon droppings (the parasite can be inhaled) and wash your hands if you touch droppings.
Raccoon droppings, which are crumbly and flat-ended, can contain a variety of food items. They are three to five inches long, but are usually broken into segments. They are about half an inch to one inch in diameter, about the size of the end of your little finger.
Raccoons defecate before climbing trees and entering structures. They create toilet areas – called "latrines" – inside and outside structures and away from the nesting area. (House cats have similar habits). You may also find scat at the base of trees, on logs and on roofs.
Raccoons make several types of noises, including a purr, a chittering sound, and various growls, snarls, and snorts.
Raccoons Too Close for Comfort
If a raccoon comes too close to you, make yourself appear larger. If you are sitting, stand up, shout and wave your arms. If necessary, throw stones or send the raccoon off with a dousing of water from a hose or bucket.
If a raccoon continues to act aggressively or strangely (circling, staggering as if drunk or disoriented) or shows unnatural tameness, it may be sick or injured. Call a game warden, your regional wildlife office, or the state police.
If aggressive raccoons are routinely seen in your area, prepare children for a possible encounter. Explain why raccoons live in the area (habitat, food sources, species adaptability) and what the children should do if one approaches. Teach them to shout a set phrase such as "Go away raccoon!" instead of simply screaming, thereby informing nearby adults of the animal's presence. Demonstrate and rehearse encounter behavior with the children.
If a raccoon finds its way into your house, stay calm, close surrounding interior doors, leave the room, and let the animal find its way back out through the open door, window or pet door. If necessary, gently use a broom to corral the raccoon outside. Do not corner a raccoon, thereby forcing it to defend itself.
A raccoon's search for food may lead it to a vegetable garden, fish pond, garbage can or chicken coop. It may find a den in an attic, chimney, or crawl space. The most effective way to prevent conflicts is to modify the habitat around your home to make it unattractive to raccoons.
Don't feed raccoons. Feeding raccoons may create an undesirable situation for your family, neighbors, pets and the raccoons themselves. Human-fed raccoons often lose their fear of people and may become aggressive when they do not receive handouts as expected. Feeding also encourages raccoons to concentrate in a small area; overcrowding can spread diseases and parasites. Finally, these hungry visitors might approach a neighbor who does not share your appreciation of the animals. The neighbor might choose to remove these raccoons, or have them removed.
Prevent raccoons from gaining access to your garbage. Keep your garbage can lid on tight by securing it with rope, chain, bungee cords or weights. Better yet, buy garbage cans with clamps or other mechanisms that hold lids on. To prevent tipping, secure side handles to metal or wooden stakes driven into the ground. Or, keep your cans in tight-fitting bins, a shed or a garage. Put garbage cans out for pickup in the morning, after raccoons have returned to their resting areas.
Feed dogs and cats indoors and keep them in at night. If you must feed your pets outside, do so in late morning or around noon, and pick up food, water bowls, leftovers and spilled food daily well before dark.
Keep pets indoors at night. If cornered, raccoons may attack dogs and cats. Bite wounds from raccoons can cause fractures and transmit disease.
Prevent raccoons from entering pet doors. Lock the pet door at night. If it is necessary to have it remain open, put an electronically activated opener on your pet's collar.
Note: Floodlights or motion detector lights placed above the pet door to scare raccoons are not long-term solutions. Keep indoor pet food and any other food away from a pet door.
Put food in secure compost containers and clean up barbecue areas. Do not put food of any kind in an open compost pile; instead, use a securely covered compost structure or a commercially available raccoon-proof composter. A covered worm box also works. Your goal is to prevent attracting raccoons and to keep yourself from being exposed to their disease-carrying droppings.
**Clean barbecue grills and grease traps thoroughly following each use.
Prevent damage to lawns. Raccoons (and skunks) are attracted to the grubs and worms that live beneath sod. For more information about preventing damage, go to "Skunks" on the main menu.
Eliminate access to denning sites. Raccoons commonly use chimneys, attics and spaces under houses, porches and sheds as den sites. Close any potential entries with one-quarter-inch mesh hardware cloth, boards or metal flashing. Make all connections flush and secure to keep mice, rats and other mammals out. Make sure you don't trap an animal inside when you seal off a potential entry. For information on securing chimneys, see below.
Figure 3: Drawing Credit - Jennifer Rees
Figure 4: Drawing Credit - Jennifer Rees
Prevent raccoons from accessing rooftops by trimming nearby tree limbs and by attaching sheets of metal flashing around corners of buildings. (Fig. 3) Farm supply centers and bird-control supply companies on the Internet often carry commercial products that prevent climbing. (Fig. 4) Remove vegetation on buildings, such as English ivy, that allows raccoons to climb walls. Hide or close the opening through which they crawl into the building.
Eliminate access to rooftops by installing sheets of aluminum flashing that are at least three feet square around the corners of buildings. (Fig. 3)
Commercially available metal or plastic spikes can help keep raccoons off of buildings. (Fig. 4)
Raccoons in Dumpsters and Down Chimneys
Raccoons are enticed by the food smells in dumpsters. When the lids are open they climb in and can't climb the slippery sides to get out. To help them escape, put a strong branch or board in the dumpster. If your disposal company leaves dumpster lids open, install a sign telling employees that it's vital to keep the lid closed so animals do not get trapped inside. Consider installing a totally enclosed trash-compacting dumpster. (You deposit your trash in the front; the trash is regularly compacted)
In spring and summer, a female raccoon may be enticed into the dark, quiet and secure environment of your chimney to nest. If you hear a large animal on the roof, or growls and whines coming from the chimney at night, there is probably a raccoon family inside. Using a powerful flashlight during the day, check whether animals have taken up residence. If spider webs are strung across the inside, you can be reasonably sure that no animal is using the chimney.
After eight to ten weeks the female and young will leave and not return. The easiest solution is to wait for the raccoons to move out on their own.
If you need to evict the animals, do not smoke them out and do not pour anything, including naphtha flakes or mothballs, down the chimney. Adult raccoons can easily climb out of a chimney, but the concentrated vapors can make the female extremely agitated while it attempts to flee. Baby raccoons cannot climb, so these measures will not evict them; in addition, the strong vapors can damage the mucous membranes of the infants.
Instead, harass the adult female using the following methods until being it is no longer worth her effort to stay. One by one, she will pick up each young animal in her mouth, latching on to the back of its neck, and move it to an alternate den. Note: Any time you try to evict any mother animal, there is a chance that she may leave some or all of the young behind.
To encourage the female raccoon to leave:
Figure 5: Drawing Credit - Jennifer Rees
To make sure the eviction process was successful, shine a powerful flashlight down the chimney during the day. Tap the chimney with a hard object and listen for any sounds of movement. If a young raccoon is left behind, it may be that the mother has abandoned it. In these rare cases it is best to hire a wildlife damage control company to remove the animal.
In urban areas, harassment techniques may not work because raccoons have become familiar with humans. If this is the case, call a wildlife damage control company and have them assess the situation.
Once the raccoons are gone, promptly call a professional chimney sweep to remove any debris and to install a commercially designed and engineered chimney cap. (Homemade caps are often unsafe and may be a fire hazard) The new cap will allow you to have fires in your fireplace or wood stove, but will keep raccoons and other wildlife from entering. (Fig. 5)
A commercial chimney cap will prevent raccoons and other small animals from entering the chimney. (Fig. 5)
Enclose poultry in a secure outdoor pen and house. Raccoons will eat chickens, ducks and turkeys and their eggs. Signs of raccoon predation include the birds' heads bitten off and left some distance away, only the bird's crop being eaten, stuck birds pulled half-way through a fence, and nests in severe disarray. Note: Coyotes, foxes, skunks, raccoons, feral cats, dogs, bobcats, opossums, weasels, hawks and owls will also prey on poultry.
If a dead bird is found with no apparent injuries, skinning it may determine what killed it. If the carcass is patterned by red spots where pointed teeth have bruised the flesh but not broken the skin, the bird was probably "played with" by one or more dogs until it died.
To prevent raccoons and other animals from accessing birds in their night roosts, equip the poultry house with a well-fitting door and a secure locking mechanism. A raccoon's dexterous paws make it possible for it to open various types of fasteners and latches.
To prevent raccoons and other animals from accessing poultry during the day, completely enclose outdoor pens with one-inch chicken wire placed over a sturdy wooden framework. Overlap and securely wire all seams on top to prevent raccoons from forcing their way in by using their weight and claws. To prevent raccoons from reaching in at ground level, surround the bottom 18 inches of the pen with smaller-mesh wire. (See "Preventing Conflicts with Skunks" for strategies to prevent raccoons from digging into enclosures)
Figure 6: Drawing Credit - Jennifer Rees
Fence orchards and vegetable gardens. Raccoons can easily climb wood or wire fences, or bypass them by using overhanging limbs of trees or shrubs. (See Figs. 6 and 8 for examples of ways to prevent raccoons from climbing fences and accessing crops at ground level) Wire fences will need to have a mesh size that is no wider than three inches to keep young raccoons out.
Install electrified wires 12 and 18 inches above ground on existing fence posts, poultry pen supports, and other structures, using the proper insulators. A single strand of wire may be sufficient, but two wires will provide added insurance that the animal will not climb up the post. Run one or two electrified wires toward the top of the fence to prevent other species from jumping the lower hot wires. (Fig. 6)
Protect fruit trees, bird feeders, and nest boxes. To prevent raccoons from climbing trees, poles, and other vertical structures, install a metal or heavy plastic barrier. (Fig. 7) Twenty-four-inch long aluminum or galvanized vent-pipe, available at most hardware stores, can serve as a barrier around a narrow support.
Note: Raccoons will attempt to use surrounding trees or structures as an avenue to access the area above the barrier.
Alternatively, a funnel-shaped piece of aluminum flashing can be fitted around the tree or other vertical structure. The outside edge of the flared metal should be a minimum of 18 inches away from the support. Cut the material with tin snips and file down any sharp edges.
Figure 7a: Drawing Credit - Jennifer Rees
Figure 7b: Drawing Credit - Jennifer Rees
Regularly pick up fallen fruit to prevent attracting raccoons.
To prevent raccoons from climbing, secure guard around trees, pipes, posts and other structures. The guard can be made from a piece of aluminum flashing or sheet metal held together with wire, nails or screws, and then painted to blend in. (Fig. 7)
Discourage raccoons from disturbing pond plants and other aquatic life. Raccoons are attracted to ponds because ponds are a source of food. Although it is tempting to simply install a motion-activated light or sprinkler – or shout at the animal when you see it – these tactics are at best temporarily effective. A raccoon, especially an urban raccoon, may run away the first night and walk away the second night. If there is no additional deterrent, however, by the third or fourth night the animal will be back even as the light shines brightly or the sprinkler sends out strong sprays of water.
To deter the animal, you must protect potential food or secure the pond itself:
Construct hiding places for fish by placing cinder blocks, ceramic drain tile, wire baskets, or upside-down plastic crates held in place with heavy rocks on the bottom of the pond.
To prevent raccoons from disturbing aquatic plants in containers, use containers that are too heavy or wide for raccoons to overturn. Securing chicken wire over the top of the containers to prevent raccoons from disturbing the soil inside.
Small ponds can be completely covered with a barrier that can be left on permanently or removed daily. Since raccoons are nocturnal, be sure the pond is covered at night. Examples of barriers include one-inch mesh chicken wire laid over the surface and held in place with stakes – raccoons will walk on the barrier and try and go under it. (While black bird-netting is less conspicuous, raccoons and other animals can easily get entangled in it) A wooden or PVC pipe frame covered with wire mesh can also be built to cover the pond. Maneuvering over pond plants with any of the above can be difficult.
Or, you can construct a frame from heavy plastic lattice available from home improvement centers. Carefully cut the lattice so it fits in the pond; cut out pieces to accommodate any pond plants. Cover the lattice with bird netting. (with the solid backing, animals are less likely to become entangled in the netting) The netting can be glued to the lattice using Shoe Goo® or other waterproof glue.
For larger ponds, stake two-foot wide strips of chicken wire flat around the inside of the pond edge where raccoons are entering. Cut the wire as needed to match the curvature of the pond. Raccoons will have difficulty reaching over the wire, and will hesitate to stand on it because of its instability. To camouflage and extend the life of the wire, spray it with dark-colored automobile undercoat paint or other rustproof paint.
Ponds with steep, two-foot high side walls discourage raccoons from entering the water, but may be a safety hazard for small children and the elderly. These hazardous areas can be located away from paths and/or be heavily buffered with dense growths of tall marginal plants and shrubs.
Figure 8: Drawing Credit - Jennifer Rees
Two electrified wires, six and 12 inches above ground and just back from the water's edge will deter raccoons. A single strand of wire may be sufficient, but two wires will provide added insurance against the animal making the climb. The wires can be hooked up to a switch for discretionary use; when you want to work near the wire, turn the system off. Where the barrier presents a safety problem, attach signs, short pieces of white cloth, or other material on the wire for visibility.
Install two electrified wires, six and 12 inches above ground around field crops and other areas needing protection. The fence can be hooked up to a switch for discretionary use; when you want to work near it, turn the system off. Where the fence presents a safety problem, install signs, short pieces of white cloth, or other material on the wire for visibility. (Fig. 8)
Trapping and relocating a raccoon several miles away seems an appealing method of resolving a conflict because it is perceived as giving the "problem animal" a second chance in a new home. Unfortunately, the reality of the situation is quite different. Raccoons typically try to return to their original territories, often getting hit by a car or killed by a predator in the process. If they remain in the new area, they may get into fights (often to the death) with resident raccoons for limited food, shelter, or nesting sites. Raccoons may also transmit diseases to rural populations that they have picked up from urban pets. Finally, if a place "in the wild" or an urban green space is perfect for raccoons, raccoons are probably already there. It isn't fair to the animals already living there to release another competitor into their home range.
Raccoons accustomed to a particular food source, type of shelter, or human activity will seek out familiar situations and surroundings. People, organizations or agencies that illegally move raccoons should be willing to assume liability for any damages or injuries caused by these animals. Precisely for these reasons, raccoons posing a threat to human and pet safety should not be relocated.
In many cases, moving raccoons will not solve the original problem because other raccoons will replace them and cause similar conflicts. Hence, it is more effective to make the site less attractive to raccoons than it is to routinely trap them.
Trapping also may not be legal in some urban areas; check with local authorities. Transporting animals without the proper permit is also unlawful in most cases.
Lethal control is a last resort and cannot be justified without first applying the above-described non-lethal control techniques. Lethal control is rarely a long-term solution as other raccoons are likely to move in if food if attractive food items such as garbage and pet foods are not eliminated or secured at the site.
If all efforts to dissuade a problem raccoon fail, the animal may have to be trapped. See Trapping Wildlife for information on trapping raccoons.
While shooting can be effective in eliminating a single raccoon, it is generally limited to rural situations. Shooting is considered too hazardous in more populated areas, even when legal.
Public Health Concerns
Canine distemper contributes significantly to raccoon mortality. It is also fatal to domestic dogs, foxes, coyotes, mink, otters, weasels and skunks. It is caused by a virus and is spread most often when animals come in contact with the bodily secretions of animals infected with the disease. Gloves, cages, and other objects that have come in contact with infected animals can also contain the virus. The best prevention against canine distemper is to have your dogs vaccinated and kept away from raccoons.
Raccoons in Maine often have roundworms (like domestic dogs and cats do, but from a different worm). Raccoon roundworm does not usually cause a serious problem for raccoons, but roundworm eggs shed in droppings can cause mild to serious illness in other animals and humans. Although rarely documented anywhere in the United States, raccoon roundworm can infect a person who accidentally ingests or inhales the parasite's eggs.
Prevention consists of never touching or smelling raccoon droppings, using rubber gloves and a mask when cleaning areas (including traps) that have been occupied by raccoons, and keeping young children and pets away from areas where raccoons concentrate. If washing raccoon droppings from a roof, for example, make sure that the water doesn't splash toys, a patio, or other similar items. Routinely encourage children to wash their hands after playing outdoors and assist them in doing so. Unfortunately, raccoon roundworm eggs can remain alive in soil and other places for several months.
Raccoons can carry rabies. If someone receives a raccoon bite or scratch, immediately scrub the wound with soap and water, then flush it liberally with tap water. Contact your physician and the local health department immediately. If your pet is bitten, follow the same cleansing procedure and contact your veterinarian. If at all possible, try to recover the animal or note where it goes, as it should be submitted to the Department of Health for rabies testing.
In addition, as previously noted, raccoon droppings may carry a parasite that can be fatal to humans. Do not handle or smell raccoon droppings (the parasite can be inhaled) and wash your hands if you touch droppings.
The raccoon is classified as both a furbearer and a game animal, and a hunting or trapping license is required to hunt or trap raccoons during an open season. Because legal status, trapping restrictions, and other information about raccoons change, contact your local Inland Fisheries and Wildlife Regional Office for updates.
If a raccoon is causing damage or is a nuisance, consult Maine's laws on this subject: http://www.mainelegislature.org/legis/statutes/12/title12ch921sec0.html
New England Wildlife, Habitat, Natural History, and Distribution
Written by: Richard DeGraff, and Mariko Yamasaki
University Press of New England, 2001.
(Available from: www.upne.com)
Resolving Human-Wildlife Conflicts: The Science of Wildlife Damage Management
Written by: Michael Conover
Lewis Publishers, 2002.
Prevention and Control of Wildlife Damage
Written by: Scott E. Hygnstrom, et al.
University of Nebraska-Lincoln, Institute of Agriculture and Natural Resources, 1994.
(Available from: University of Nebraska Cooperative Extension, 202 Natural Resources Hall, Lincoln, NE 68583-0819; phone: 402-472-2188; also see Internet Sites below.)
- Wildlife Control Supplies
- U.S. Forest Service Wildlife Species Life Form Information
- Tomahawk Live Traps
Adapted from: "Living with Wildlife in the Pacific Northwest"
(see Washington Department of Fish and Wildlife)
Written by: Russell Link, Wildlife Biologist, Email Russell Link, with assistance from WDFW Biologists Rich Beausoleil and Rocky Spencer
Design and layout: Peggy Ushakoff, ITT2
Illustrations: As credited
Copyright 2005 by the Washington Department of Fish and Wildlife
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#### Please Solve R. D. Sharma class 12 Chapter 4 Algbra of Matrices Exercise Multiple Choice Questions Question 4 Maths textbook Solution.
Answer: The correct option is $\text { (B), } \mathrm{B}^{2}=\mathrm{B} \text { and } \mathrm{A}^{2}=\mathrm{A}$
Given:$A B=A, B A=B$
Solution:
$A B=A$ ....(1)
$BA=B$ ....(2)
Now from Equation (2),
\begin{aligned} &B \times(A B)=B \\ &B^{2} A=B \end{aligned}
Again from Equation (2),
\begin{aligned} &\mathrm{B}^{2} \mathrm{~A}=\mathrm{BA} \\ &\mathrm{B}^{2}=\mathrm{B} \end{aligned}
Now from Equation (1),
\begin{aligned} &A \times(B A)=A \\ &A^{2} B=A \end{aligned}
Again from Equation (1),
\begin{aligned} &\mathrm{A}^{2} \mathrm{~B}=\mathrm{AB} \\ &\mathrm{A}^{2}=\mathrm{A} \end{aligned}
So,$\mathrm{A}^{2}=\mathrm{A} \text { and } \mathrm{B}^{2}=\mathrm{B}$
Hence, the correct option is (B).
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crawl-data/CC-MAIN-2023-40/segments/1695233510358.68/warc/CC-MAIN-20230928031105-20230928061105-00366.warc.gz
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Published on in Trisomy 21 Update
Walking up and down stairs is a skill we do multiple times throughout a day. It becomes an automatic skill that we do not think about performing once we achieve the motor plan of stair negotiation. We see stairs in front of us and are able to easily walk up and down without much thought, with good balance and with one foot on each step.
Most children begin walking up and down the stairs around 2 years old, after they have refined their independent walking skills. Children with trisomy 21 can also begin walking up and down the stairs shortly after they learn to walk — with appropriate modifications and support for the task.
When a child is able to four-point crawl but is not yet walking independently, the child can creep up stairs and creep down the stairs backward on his belly. Many children are interested in creeping up the stairs, but may be resistant to coming down backward. Activities such as coming down off the couch and bed backward and sliding down a slide backward serve as practice for coming down the stairs backward. Some children are fearful of moving backward because they cannot see where they are going. Start a few stairs from the bottom at first if the child is fearful. Always be close to the child for supervision to prevent him from falling down the stairs. Use appropriate gates at the top and bottom of all stairs to block stair access when you are not able to assist your child.
Once your child is walking and ready to learn how to navigate the stairs, these are some helpful tips to help with success and efficiency and to help prevent injury.
Walking up stairs
Progressions I like to use when first teaching children how to walk up stairs are:
- Practice walking up first with two feet on each step and holding a railing. You should be behind your child on the stairs. Your child may want to push back against you for support, which will result in her straightening her leg and extending the knee instead of bending at the knee and ankle to shift weight forward on the leg that is on the higher step. Provide her with support and use of the rail and encourage her to bring one foot up to the next step, then bring the second foot up to meet it. Make sure you alternate feet when practicing to avoid only one leg achieving the skill or becoming stronger than the other leg.
- In the school setting, start by side stepping up the stairs if your child needs to keep up with his classmates. This will give him more support and speed.
- Once your child can walk up the stairs with two feet on each step, but alternating which foot leads, with use of a railing, practice placing only one foot on each step using a foot-overfoot pattern. Ask her to walk “like a big kid.”
- The next progression is to practice the stairs without a railing, which may require going back to putting two feet on each step when first learning this skill.
Walking down stairs
Progressions I use when teaching children how to walk down stairs are:
- First practice squats on the floor to pick up small toys and make sure they are comfortable bending their knees slowly with good mid-range control in standing. Children with Trisomy 21 often have a hard time controlling how fast they bend their knees. They will often stand and “lock” their knees into extension or they will bend their legs and squat all the way down to the ground without being able to hold the middle positions. In other words, they do not use their quadriceps muscles eccentrically to control how fast they bend down to pick up a toy. This skill is needed to walk down the stairs successfully.
- Have your child practice holding a railing and bringing one leg down at a time and then alternating their feet. You should be in front of him. He may need you to help him alternate his feet. Children will often want to advance their weaker leg first, while the stronger leg does the work of lowering, and may need help to advance the stronger leg. When stepping down, the leg on which you are standing is the weight-bearing leg and is the one doing all the work.
- If your child is nervous about stepping down, try side stepping with two hands on the railing or start at the bottom few stairs and then increase the number of stairs to practice as performance improves.
- Practice placing the foot on the stair and keeping the knee just a little bent. Your child should avoid coming down on a “locked knee” as this can lead to knee injury over time.
- Once your child can step down alternating feet with two feet on each step, practice taking “big kid” steps and placing only one foot on each step.
Helpful strategies to help children alternate their feet include:
- Place a different sticker (such as a Sesame Street character) on each foot and then say: “Give Elmo a turn, then give Cookie Monster a turn.” This is a helpful visual cue.
- Tape cut-out feet of different colors to the stairs, such as a red foot for the left foot and a blue foot for the right foot. Ask your child to step on the red foot then the blue foot.
- Place a strip of brightly colored duct tape along the edge of the steps if your child has any visual impairments or visual perceptual issues. This helps her identify the edges of a stair.
At school, stairways tend to be busy, loud areas. Walking without a railing, with a backpack and with noise are all challenges faced at school. Ask the school physical therapist to make sure your child is safe and successful navigating stairs. Stair skills can be practiced with the therapist in a controlled, quiet environment and followed through at home to gain the necessary practice before transferring the skill to the busy school day. Timing your child while on the stairs and comparing it to the time it takes for the rest of the class to do the stairs is important information when evaluating stairs. A child may be successful at home and in therapy but have a hard time at school.
Children will encounter stairs when they go out to the playground at recess. Walking up stairs to a slide, stepping over a small curb or railroad tie, and stepping down off a small step without a railing are common obstacles on a playground. Practicing these skills at home and in therapy may help them improve on the playground.
Your child’s footwear can also be a factor on the playground and stairs. Children with trisomy 21 often have pronated feet (when feet roll inward), low muscle tone, decreased strength and ligamentous laxity. Make sure your child has appropriate footwear, preferably a supportive tie sneaker and orthotics, if prescribed. Your child’s backpack should also be considered. Make sure the bag is the correct size, the straps are adjusted correctly and your child is not carrying too many items, making the backpack too heavy.
Stair climbing is an important skill that can be improved with practice at home, during school and in therapy. Consult with your child’s physical therapist to inquire how well your child walks up and down the stairs during the school day. Practice, proper footwear and an appropriate backpack will have an impact on your child’s success keeping up with classmates during the school day.
Contributed by: Helen Milligan, PT, MPT
Categories: Trisomy 21
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By Carl Wilson, Horticulturist, Denver Cooperative Extension
Aster yellows is a severe disease of many flowers and vegetables. It is caused by a
bacterial-like organism and is carried by the aster leafhopper. The leafhoppers annually
migrate to Colorado from overwintering areas along the Gulf of Mexico. Once insects feed
and acquire the disease organism, they remain infective for life.
Aster yellows affect some 300 species of plants including aster, marigold, zinnia,
petunia, lettuce, carrots, beets and onions. Though symptoms vary, plants usually show
yellowed, bronzed, and twisted new shoot or flower growth. The most severe symptoms are
seen in late July and August and don't let up until frost.
Control is difficult. As long as leafhoppers abound, the disease can be carried to plants.
And leafhoppers may move into gardens from many areas. The most practical remedy for
repeated disease occurrences over several seasons may be to grow less susceptible flowers.
These include nicotiana, geraniums, salvia, cockscomb, impatiens, portulaca and verbena.
Photo of Aster Yellows on Cosmos: Whitney Cranshaw
Photograph of Aster Leafhopper courtesy of University of California
Statewide Integrated Pest Management Project.
Back to Pests
Back to Home
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# Measures & Significant Figures
## I. Introduction
The following discussion should be thoroughly understood by the student before going ahead with any experiment in the course. A through knowledge of significant figures and the determination of experimental uncertainty will be of great assistance both in reporting your measurements and in evaluating your results.
The sections below are laid out in the order you will need when first learning about this material. Proceed from one section to the next working the practice exercises along the way. Afterword you can browse the topics as needed to refresh yourself on any topic.
## II. Determining the uncertainty of an experimentally deduced result.
Many experiments will have a numerical result as the outcome. The process of measurement will always involve some uncertainty and therefore the result computed from these measurements will have some uncertainty. Our goal is to be able to estimate the size of this uncertainty and report it along with the experimental result. A variety of concepts and procedures are needed for this task. They are taken up in the sections below.
### A. Significant Figures
Rounding and doing math with significant figures.
### B. Accuracy vs. Precision and Error vs. Uncertainty.
Important definitions with examples.
How do you decide the number of significant figures when you make a measurement?
### D. Mean Values
Increased confidence through averaging.
### E. The Average Deviation of the Mean.
The Average Deviation provides an uncertainty with less guesswork.
### F. Relative Uncertainty
A good way to compare uncertainties.
### G. Precision of Computed Results
What to do when the final result is computed from several measurements, each with some uncertainty.
### H. Precision of Computed Results (Formal method using Calculus)
What to do when the final result is computed from several measurements, each with some uncertainty.
### III. Comments on Sources of Uncertainties
Experimental uncertainty due to random measurement effects are distinct from systematic errors introduced by defective equipment or procedures or caused by unknown influences on the equipment. Also we will treat the uncertainty differently if it arises from an independent source than when it is related to a source already considered.
1. Random error.
The error will be different with each repetition of the measurement.
2. Systematic error.
Repeating the measurement will reproduce the same error.
3. Independent sources of error.
Uncertainty is compounded.
Last Updated August 4, 2015
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Over the past several decades, bacteria have become increasingly resistant to available drugs. One strategy that might combat such resistance would be to overwhelm bacterial defenses by using highly targeted nanoparticles to deliver large doses of existing antibiotics.
In a step toward that goal, researchers at the Institute and Brigham and Women’s Hospital have developed a nanoparticle designed to evade the immune system and home in on infection sites to unleash a focused antibiotic attack.
This approach would mitigate the side effects of some antibiotics and protect the beneficial bacteria that normally live inside our bodies, says Aleks Radovic-Moreno, an MIT graduate student and lead author of a recent paper describing the particles in the journal ACS Nano.
The team, led by Institute Professor Robert Langer of MIT and Omid Farokzhad, director of the Laboratory of Nanomedicine and Biomaterials at Brigham and Women’s, created the new nanoparticles from a polymer capped with polyethylene glycol (PEG). PEG is commonly used for drug delivery because it is nontoxic and can help nanoparticles evade detection by the immune system to travel through the bloodstream.
Their next step was to induce the particles to target bacteria. Researchers have previously tried giving drug-containing particles a positive charge, which attracts them to bacteria’s negatively charged cell walls. However, the immune system tends to clear positively charged nanoparticles from the body before they can encounter bacteria.
To overcome this obstacle, the MIT and Brigham and Women’s team designed nanoparticles that can switch their charge depending on their environment. While they circulate in the bloodstream, the particles have a slight negative charge. But when they encounter an infection site, which tends to be slightly acidic, they gain a positive charge, allowing them to bind tightly to bacteria and release their drug payload.
These particles were designed to deliver vancomycin, a common treatment for drug-resistant infections, but they could be modified to deliver other antibiotics or combinations of drugs.
Although further development is needed, the researchers hope the high doses delivered by their particles could eventually help overcome bacterial resistance. “When bacteria are drug resistant, it doesn’t mean they stop responding,” Radovic-Moreno says. “It means they respond, but only at higher concentrations. And the reason you can’t achieve these concentrations clinically is because antibiotics are sometimes toxic, or they don’t stay at that site of infection long enough.”
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To explain the difference we use two words: 'magnitude' and 'direction'. By magnitude we mean how much of the quantity is there. By direction we mean is this quantity having a direction which defines it.
Physical quantities which are completely specified by just giving out there magnitude are known as scalars. Examples of scalar quantities are distance, mass, speed, volume, density and temperature.
Other physical quantities cannot be defined by just their magnitude. To define them completely we must also specify their direction. Examples of these are velocity, displacement, acceleration, force, torque and momentum. These are called vectors.
If we were to represent the magnitude and direction of two vectors by two adjacent sides of a parallelogram. The resultant can then be represented in magnitude and direction by the diagonal. This diagonal is the one which passes through the point of intersection of these two sides.
It is often necessary to split a vector into its components. Splitting of a vector into its components is called resolution of the vector. The original vector is the resultant of these components. When the components of a vector are at right angle to each other they are called the rectangular components of a vector.
In the figure above the green vector has been resolved into two vectors: blue and red. These vectors are at right angles to each other. The are the rectangular components of the green vector.
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CAE Reading and Use of English Part 1
For questions 1-8, read the text below and decide which answer (А, В, C or D) best fits each gap. There is an example at the beginning (0).
High notes of the singing Neanderthals
Neanderthals have been misunderstood. The early humanoids traditionally 0 CHARACTERIZED as ape-like brutes were deeply emotional beings with high-pitched voices. They may 1 have sung to each other. This new image has 2 from two studies of the vocal apparatus and anatomy of the creatures that 3 Europe between 200,000 and 35,000 years ago.
The research shows that Neanderthal voices might well have produced loud, womanly and highly melodic sounds – not the roars and grunts previously 4 by most researchers. Stephen Mithen, Professor of Archaeology and author of one of the studies, said: ‘What is emerging is a picture of an intelligent and emotionally complex creature whose most likely 5 of communication would have been part language and part song.’
Mithen’s work 6 with the first detailed study of a reconstructed Neanderthal skeleton. Anthropologists brought together bones and casts from several sites to re-create the creature. The creature that emerges would have 7 markedly from humans, Neanderthals seem to have had an extremely powerful 8 and no waist.
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Medieval period is a long-lasting era of feudalism. In Europe it lasted for 12 years, in many Asian countries — even longer. But elements of Medieval feudalism have not disappeared completely and still remain in certain countries. In every country feudalism became a symbol of peasant’s submission to landowners. During the Early Medieval Era in Europe there was gradually formed civilization, which influenced further world development. Feudalism converted land (the main field of human activity) into monopolized property which included state and private one. Feudalism is considered to be a progress in human development. Peasants were interested in the growing labour-productivity. Manufacture and new classes of Bourgeoisie began to appear gradually. During the Medieval period ancient tribes converted into nationalities. It leads to changes in society and further formation of modern nations. Feudal society is characterized by class struggle between exploiting and exploitable people. It was the main reason of rebellions. Though common people were always defeated, they managed to get over feudalism with the help of bourgeoisie. On the whole the feudal period in Europe is known for its political fragmentation.
There are several periods in history of Medieval era. Each of them contains certain changes in economical, social and political spheres. Many countries became feudal without passing through primitive communal system. Others passed through all stages of development.
- formation of feudal social classes (landowners and dependent peasants)
- peak of feudalism (fragmentation among state authorities)
- centralization of state power; prosperity of city culture; appearance of humanism as new ideology; reformation of catholic church
- sharpening of feudal antagonism (contradictions); formation of capitalism
Many phenomena in the life of modern nations and states came from Medieval era — formation of bourgeois society, development of national cultures, revolutionary struggle of oppressed classes, struggle for free-thinking. The research in the sphere of Medieval history helps to understand present situation in society and to predict perspectives for future development.
Spreading of Christianity in the Early Medieval period lead to appearance of cathedrals, monasteries and convents. Earliest institutions of this kind were formed due to state investitures.
Meroving Realm. All the epoch of 6-7 centuries is usually called the period of Great migration of nations. Actually at that time lots of people left their motherlands, where they’d been living for hundreds of years, and went to conquer new lands. European map changed beyond recognition. Numerous invasions wiped Roman Empire off the map. It was replaced by German realms. Rome and the whole antic world were destroyed. In this way Europe entered Medieval Era. This period is famous for dreadful bloody battles. Famous FRANC tribes, who are considered to be skillful in military affairs, participated in many of them. Enemies were particularly afraid of francs’ battle- and pole-axes, which were used with unbelievable strength and great accuracy. At first francs lived along the lower reach of Rein near Gallian frontier. All francs used to cut their hair and beards, unlike other tribes. Only members of Royal family were allowed to have long hair. In the end of the 5th century all franc tribes were united under the command of Hlodwig (Meroving by origin). Acting with slyness and cruelty, he managed to remove all other franc leaders and began to rule alone. He gained respect among his people due to amazing luck in battles — he managed to defeat Roman governor in the year of 468 and to create his own realm. Royal dynasty of Hlodwig is known as the dynasty of Merovings in honor of Hlodwig’s legendary ancestor. Merovings ruled their realm till the middle of the 8th century.
Franc realm adopted Roman monetary system, according to which the basic coin was called golden solid. Treasury was usually enriched at the expense of military loot. Later in addition to golden coins appeared silver ones.
Way of Hlodwig from an ordinary leader to the King was not so easy as it may seem to be. He met with resistance of his people and tried to cope with it. Realizing that Roman church could be an irreplaceable ally, Hlodwig was first among other barbarian leaders to adopt Christianity per Roman sample. He gave up paganism and was christened together with his fighting squads. Later all francs followed his example. Many francs were not satisfied with Hlodwig’s decision, but were just afraid of him. Instead, Romans were very pleased. Historians doubt that Hlodwig knew much about Christian doctrine, but his action occurred to be very wise. From that time on Francs and Romans began mix into one nation — they were no longer separated by religion and francs began to join many spheres of Roman culture. But Hlodwig suspected another privilege of new religion as well. Christianity says that any power comes from God. Consequently it means that it also concerns Hlodwig and his heirs. In that way adoption of Christian religion strengthened Hlodwig’s power, raising him above his people.
Either Hlodwig or some of his descendants ordered to write down ancient laws of francs. To put it more precisely they were more customs than laws actually. Nobody was allowed to make up new laws. They could only follow ancient traditions. At that time people thought the only right rules were those of ancient origin. Any innovations can only do harm. Elderly people imparted their knowledge to children and grandchildren. But nevertheless people somehow changed ancient order usually when they were not satisfied with certain tradition.
So-called “Salic truth”, a summary of ancient customs and norms, made by order of Hlodwig became the most outstanding example of ancient law in Europe. The name of the document comes from the name of franc tribe — Salic francs. The reason for creation of such a summary was Hlodwig’s desire to be a chief judge himself. In case something in “Salic truth” is not clear, people are recommended to consult the King. But he could explain the law according to his own interests. Moreover it was King who decided whether some law should be included into “Salic truth” or not. This document shows the strengthening of the King’s power. He presses towards becoming a true sovereign.
After the death of Hlodwig each of his four sons inherited some part of the realm. Conflicts, following this division, lead to decadence of Meroving state.
Karoling Empire. In the 8th century collapse in Franc realm not only paused but quite the contrary united again. Francs restored their power over former territories and even moved forward joining Italy, major part of Germany and Northern Spain. But how could they manage to do that? Turning point in the history of Franc state is change of Royal dynasty. Meroving Kings were replaced by KAROLINGS, who belonged to a noble dynasty of Eastern francs. They gained the highest administrative position of MAJORDOMES. It meant the position of superior person, who runs the economy of the palace. Later he was given rights to manage Royal property (land and treasury) in the whole country. The King’s power became dependant on Majordome’s wealth and authority. Representatives of Meroving dynasties still considered themselves Kings even though they could no longer influence life in their realm. Contemporaries disdainfully called them “lazy kings”. Long-haired “lazy kings” were deprived of nearly all power and nobody paid attention to them anymore.
Authority of Karolings abruptly strengthened during the rein of majordome Karl Martell. He gained respect among people, appointing them higher positions. As a result his army was reinforced with more people. Karl gained lasting power and ascendancy in many lands. His son Pipin the Short continued his father’s policy and made another step on the way to strengthening the power of his family. He officially declared himself the King of francs, while the last Meroving king became a monarch. The whole territory from La-Manche to the coast of the Mediterranean belonged to Pipin the Short. But the peak of development was achieved during the rein of his son Karl the Great, who managed to conquer major part of Western Europe and Emperor with the help of the Pope.
The history of Karoling rein shows that reunion development of the Franc state happened owing to the representatives of new dynasty. Their reforms were of great importance.
Karl the Great is usually depicted as a powerful monarch, enlightener, founder of Christian state, who united numerous European nations. Under the will of Pipin the Short, the state was divided between Karl and his younger brother Karloman. Brothers’ quarrel nearly lead to war, but unexpected death of Karloman gave Karl an opportunity to unite all lands under his rein. Karl organized military campaigns nearly every year. The longest and the most painful were lead in the East.
The most famous campaigns were lead against Spanish moors. On the way back Karl was attacked by duke Lopus (Gascon) but evaded death owing to his commander Roland. Roland hampered the attackers till late night and in this way gave Karl an opportunity to escape. Poland perished together with his troop. Next year Karl destroyed Gascon duchy and executed duke Lopus. All this was vividly described in “Story of Roland” — one of the most prominent samples of medieval epos. Roland became a symbol of military valor.
Under the arrangement of Karl the Great all ancient regulations concerning orders in social and military service were improved and systematized. These regulations precisely defined people and their duties in military affairs. In the time of Karl new military equipment such as long shields, big bows, breast armor, helmets and mails were introduced. The number of soldiers in cavalry was increased and became nearly equal to infantry. All people were obliged to support armies with certain quantity of bread, provisions, horses and beasts of burden.
Karl’s vast activity in the field of education was dedicated to Christian upbringing. From his youth up Karl respected enlightenment, even though he remained illiterate for long time. He created a decree, according to which all children of free people ought to be educated. But this couldn’t be fulfilled at that time.
In private life Karl remained German King: he wore national clothes — shirt, trousers and a coat. Only in Rome he dressed in the clothes of Roman patrician as a token of respect to the Pope. According to the same German custom, he had several wives; one of them was considered the head one, though it contradicted with Christian laws. Out of his three legal sons — Karl, Pipin and Ludovik — only the youngest won lived longer that his father. It is also said that love to his daughters was so great, that he didn’t want them to get married.
Karl the Great was a really remarkable ruler and governor. His main merit is that he used political achievements of forerunners to full extent. An important element is also certain connection between Karl’s Empire and the formation of three great European states — France, Germany and Italy.
On the whole Early Medieval period is characterized by prosperity in different spheres of human’s life which lead to further developments and formation of civilization.
CARRIE, a full-text electronic library, created at the University of Kansas World Wide Web Virtual Library: History www.iue.it/VL/history/index.html – 53k
Manual for school-pupils “History of Medieval Ages” — Moscow, 2001.
Popo de Bolier “Medieval France” — Moscow, 2000.
Sinova I. “History of Medieval Era” — publishing house Litera, 2002
ATTENTION! Another set of tips didn’t work for your particular situation? That happens more often than you can imagine, but we have got you covered. At EssayLib.com you can get a customized research paper on Medieval Period written in strict accordance with your professor’s instructions. Just fill out the inquiry form and get to know the price of your order, the writers available and more details about the service. You pay nothing at this stage, so why not to try?
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Definition: Given two matrices $A$ and $B$, both of which are of size $m \times n$, then the sum denoted $A + B$ is an $m \times n$ matrix whose entries are formed by adding corresponding entries of $B$ to corresponding entries of $A$. If $C = A + B$, then $c_{ij} = a_{ij} + b_{ij}$. If the size of matrix $A$ and matrix $B$ are not the same size, then the sum $A + B$ is said to be undefined.
Let's first look at the following $2 \times 3$ matrices $A$ and $B$:
(1)
\begin{align} A = \begin{bmatrix} 3 & 0 & 2\\ 1 & 4 & 2 \end{bmatrix} \quad , \quad B = \begin{bmatrix} 2 & 1 & 3\\ -2 & 5 & 10 \end{bmatrix} \end{align}
To determine the sum of matrix both matrices ($A + B$), we will add corresponding entries of $A$ to $B$. For example, to determine the first entry in our sum, we will take $a_{11} + b_{11}$, that is $3 + 2 = 5$:
(2)
\begin{align} \quad A + B = \begin{bmatrix} 3 & 0 & 2\\ 1 & 4 & 2 \end{bmatrix} + \begin{bmatrix} 2 & 1 & 3\\ -2 & 5 & 10 \end{bmatrix} = \begin{bmatrix} 3 + 2 & 0 + 1 & 2 + 3\\ 1 + (-2) & 4 + 5 & 2 + 10 \end{bmatrix} = \begin{bmatrix} 5 & 1& 5\\ -1 & 9 & 12 \end{bmatrix} \end{align}
Therefore we have that:
(3)
\begin{align} A + B = \begin{bmatrix} 5 & 1& 5\\ -1 & 9 & 12 \end{bmatrix} \end{align}
In general, if we have two $m \times n$ matrices $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & & a_{2n}\\ \vdots & & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$ and $B = \begin{bmatrix}b_{11} & b_{12} & \cdots & b_{1n}\\ b_{21} & b_{22} & & b_{2n}\\ \vdots & & \ddots & \vdots\\ b_{m1} & b_{m2} & \cdots & b_{mn} \end{bmatrix}$, then the sum $A + B$ is as follows:
(4)
\begin{align} \quad A + B = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & & a_{2n}\\ \vdots & & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} + \begin{bmatrix}b_{11} & b_{12} & \cdots & b_{1n}\\ b_{21} & b_{22} & & b_{2n}\\ \vdots & & \ddots & \vdots\\ b_{m1} & b_{m2} & \cdots & b_{mn} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12}& \cdots & a_{1n} + b_{1n}\\ a_{21} + b_{21}& a_{22} + b_{22}& & a_{2n} + b_{2n}\\ \vdots & & \ddots & \vdots\\ a_{m1} + b_{m1} & a_{m2} + b_{m2}& \cdots & a_{mn} + b_{mn} \end{bmatrix} \end{align}
## Example 1
Given the following matrices, determine the resulting matrix $A + B$:
(5)
\begin{align} A = \begin{bmatrix} 2 & 1\\ 4 & 3 \end{bmatrix} \quad , \quad B = \begin{bmatrix} 9 & -2\\ -5 & -3 \end{bmatrix} \end{align}
We must first sum up corresponding entries:
(6)
\begin{align} a_{11} + b_{11} = 2 + 9 = 11 \\ a_{12} + b_{12} = 1 + (-2) = -1 \\ a_{21} + b_{21} = 4 + (-5) = -1 \\ a_{22} + b_{22} = 3 + (-3) = 0 \end{align}
These are the entries of our matrix and therefore:
(7)
\begin{align} A + B = \begin{bmatrix} 11 & -1\\ -1 & 0 \end{bmatrix} \end{align}
# Matrix Subtraction
Definition: Given two matrices $A$ and $B$, both of which are of size $m \times n$, the difference $A - B$ is an $m \times n$ matrix whose entries are formed by subtracting entries of $B$ from corresponding entries of $A$. If $C = A - B$, then $c_{ij} = a_{ij} - b_{ij}$. If the size of matrix $A$ and matrix $B$ are not the same, then the difference $A - B$ is said to be undefined.
Subtracting two same-size matrices is very similar to adding matrices with the only difference being subtracting corresponding entries.
## Example 2
Using the matrices from example 1, determine the resulting matrix $A - B$.
This time we will find the difference between the entries of $B$ from $A$ (taking an entry of $A$ and subtracting the corresponding entry in $B$):
(8)
\begin{align} a_{11} - b_{11} = 2 - 9 = -7 \\ a_{12} - b_{12} = 1 - (-2) = 3 \\ a_{21} - b_{21} = 4 - (-5) = 9 \\ a_{22} - b_{22} = 3 - (-3) = 6 \end{align}
These are the entries of our matrix and therefore:
(9)
\begin{align} A - B = \begin{bmatrix} -7 & 3\\ 9 & 6 \end{bmatrix} \end{align}
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The nectarine tree (P. persica nectarina), a peach variant, belongs to the prunus family of plants. Its fruit, the rich, smooth textured nectarine, is smaller, redder and more aromatic than the peach. Nectarine trees are relatively short and reach between 10 and 30 feet in height. They prefer rich, well-drained soils and plenty of water in the summer months, and prefer the conditions found in U.S. Department of Agriculture zones 6 to 9. The trees, however, are fussy, and improper care makes them susceptible to a host of pests, diseases and conditions.
Nectarine trees are vulnerable to bacterial canker disease -- their biggest killer. The disease, caused by the bacterial pathogen Pseudomonas syringae, infects fruit, leaves, blossoms, trunk and branches of the tree. Signs of infection include a waxy gum exuded from infected parts of the trunk and spots on young stems and branches. Severely infected trees wilt and die. Bacterial canker is more prevalent during wet seasons and early winter. Manage canker by avoiding pruning in early winter. Cut and remove all infected parts and burn them immediately. Apply a copper-based fungicide spray, which will effectively control the disease.
Bacterial spot, caused by the bacterium Xanthomonas campestris pv. Vesicatoria, damages foliage, fruit and shoots. It causes defoliation, lesions on developing fruit and blossom blight. Infected leaves discolor and drop off. Severe defoliation results in smaller fruit size, surface cracks and sunburns. Bacterial spot disease, once set in, is difficult to control. Apply oxytetracycline and copper-based fungicides for preventive control for the next year. Invest in bacterial spot resistant nectarine tree variants, such as "Emeraude," "Stark Ovation" and "Fantasia."
Insect pests common to the nectarine tree include peach twig borers, aphids, spider mites and Western flower thrips. Peach twig borers feed on and damage nectarine fruit. Aphids attack shoot tips, causing malformation and stunting. Spider mites attack leaves, depriving them of chlorophyll — the green pigment the plant uses to manufacture food. Western flower thrips feed on and damage young nectarine fruit and flowers. Insecticides including malathion, carbaryl and spinosad control peach twig borers. Insecticidal soap suppresses aphids during the growing season while 2-percent oil spray kills aphid eggs. Horticultural oil controls spider mites. Application of the insecticide spinosad prevents Western flower thrips.
Appropriate care prevents nectarine tree problems. Keep your trees healthy. Old and weak trees are more susceptible to disease and insect attack. Provide trees ample sunlight and well-drained soil rich in organic compost. Prune them to allow sunlight to reach all branches. Inspect them frequently for signs of disease and pests. Contact your local cooperative extension to suggest proper treatment if you suspect a problem.
- Jupiterimages/liquidlibrary/Getty Images
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Print out these free letter practice worksheets to help your kids learn to recognize and write letters and the alphabet, in both lower and upper cases. Check out our comprehensive collection of printables for teaching preschool and kindergarten children the alphabet. Whether you are a teacher, homeschooling your children or a parent, these free alphabet worksheets are perfect for helping kids learn their ABC’s. Check out the following alphabet worksheets that we have on the list below.
On this page, you will find a large assortment of various letter practice worksheets that are all free to print. These alphabet worksheets include some basic alphabet tracing. Whether you are a classroom teacher or a parent teaching kids to write at home, you’ll find plenty of great worksheets here. All you need is only choosing the worksheets, and then get to print it right from your device. Browse more of the letter worksheets in the images below.
Printable writing materials are important for preschool, kindergarten and early elementary. Learn to write with sequenced numbered arrows and dotted guidelines on the following alphabet worksheets. These are suitable for older toddlers, preschool, kindergarten and first grade.
Using the letter worksheets children will learn how to write alphabet by tracing and writing every letter. You can introduce these worksheets once your kids can comfortably trace each letter on letter tracing worksheets. Download the worksheets right from your device and get started right away! Don’t forget to visit again for more worksheets and templates!
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# Search by Topic
#### Resources tagged with Factors and multiples similar to HCF Expression:
Filter by: Content type:
Stage:
Challenge level:
### There are 93 results
Broad Topics > Numbers and the Number System > Factors and multiples
### Stars
##### Stage: 3 Challenge Level:
Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?
### Inclusion Exclusion
##### Stage: 3 Challenge Level:
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
### Sieve of Eratosthenes
##### Stage: 3 Challenge Level:
Follow this recipe for sieving numbers and see what interesting patterns emerge.
### Different by One
##### Stage: 4 Challenge Level:
Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)
### N000ughty Thoughts
##### Stage: 4 Challenge Level:
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
### Factoring Factorials
##### Stage: 3 Challenge Level:
Find the highest power of 11 that will divide into 1000! exactly.
### There's Always One Isn't There
##### Stage: 4 Challenge Level:
Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.
### Powerful Factorial
##### Stage: 3 Challenge Level:
6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?
### Power Crazy
##### Stage: 3 Challenge Level:
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
### Napier's Location Arithmetic
##### Stage: 4 Challenge Level:
Have you seen this way of doing multiplication ?
### Substitution Cipher
##### Stage: 3 and 4 Challenge Level:
Find the frequency distribution for ordinary English, and use it to help you crack the code.
### Factors and Multiples Game
##### Stage: 2, 3 and 4 Challenge Level:
A game in which players take it in turns to choose a number. Can you block your opponent?
### Counting Cogs
##### Stage: 2 and 3 Challenge Level:
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
### A First Product Sudoku
##### Stage: 3 Challenge Level:
Given the products of adjacent cells, can you complete this Sudoku?
### Hidden Squares
##### Stage: 3 Challenge Level:
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
### Big Powers
##### Stage: 3 and 4 Challenge Level:
Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.
### Substitution Transposed
##### Stage: 3 and 4 Challenge Level:
Substitution and Transposition all in one! How fiendish can these codes get?
### Factorial
##### Stage: 4 Challenge Level:
How many zeros are there at the end of the number which is the product of first hundred positive integers?
### Missing Multipliers
##### Stage: 3 Challenge Level:
What is the smallest number of answers you need to reveal in order to work out the missing headers?
### Factors and Multiples - Secondary Resources
##### Stage: 3 and 4 Challenge Level:
A collection of resources to support work on Factors and Multiples at Secondary level.
### Three Times Seven
##### Stage: 3 Challenge Level:
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
### Factor Track
##### Stage: 2 and 3 Challenge Level:
Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.
### Charlie's Delightful Machine
##### Stage: 3 and 4 Challenge Level:
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
### Data Chunks
##### Stage: 4 Challenge Level:
Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .
### Multiplication Equation Sudoku
##### Stage: 4 and 5 Challenge Level:
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
### LCM Sudoku II
##### Stage: 3, 4 and 5 Challenge Level:
You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.
### Number Rules - OK
##### Stage: 4 Challenge Level:
Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
### Transposition Cipher
##### Stage: 3 and 4 Challenge Level:
Can you work out what size grid you need to read our secret message?
### The Remainders Game
##### Stage: 2 and 3 Challenge Level:
A game that tests your understanding of remainders.
### Got It
##### Stage: 2 and 3 Challenge Level:
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
### Thirty Six Exactly
##### Stage: 3 Challenge Level:
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
### Product Sudoku 2
##### Stage: 3 and 4 Challenge Level:
Given the products of diagonally opposite cells - can you complete this Sudoku?
### Gaxinta
##### Stage: 3 Challenge Level:
A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?
### Ewa's Eggs
##### Stage: 3 Challenge Level:
I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?
### How Old Are the Children?
##### Stage: 3 Challenge Level:
A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"
##### Stage: 3 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?
### LCM Sudoku
##### Stage: 4 Challenge Level:
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
### Diggits
##### Stage: 3 Challenge Level:
Can you find what the last two digits of the number $4^{1999}$ are?
### Factors and Multiples Puzzle
##### Stage: 3 Challenge Level:
Using your knowledge of the properties of numbers, can you fill all the squares on the board?
### A Biggy
##### Stage: 4 Challenge Level:
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
### Common Divisor
##### Stage: 4 Challenge Level:
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
### GOT IT Now
##### Stage: 2 and 3 Challenge Level:
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
### Squaresearch
##### Stage: 4 Challenge Level:
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
### Repeaters
##### Stage: 3 Challenge Level:
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
### One to Eight
##### Stage: 3 Challenge Level:
Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.
### Special Sums and Products
##### Stage: 3 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.
### Factoring a Million
##### Stage: 4 Challenge Level:
In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
### Take Three from Five
##### Stage: 3 and 4 Challenge Level:
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
### Shifting Times Tables
##### Stage: 3 Challenge Level:
Can you find a way to identify times tables after they have been shifted up?
### Mathematical Swimmer
##### Stage: 3 Challenge Level:
Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .
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פֿ Online ᆎ Beaks! for kids ᐥ By Sneed B Collard III ᓃ Young naturalists explore a variety of birds, their habitats, and how their beaks help them build, eat, and survive From the twisted beak of a crossbill to the color changing bill of a seagull, readers will learn fun facts about how beaks are designed and used as tools by birds of all shapes and sizes Bright, bold cut paper illustrations create amazingly realistic tableaus of birds in their natural environments with their beaks in action Back matter includes a comprehensive quiz, a bibliography, and a list of related websites. Bird Beaks Backyard Nature A brief introduction to beaks Here are the main bird beak, or bill, types Short, thick, curved, pointed of hawks, falcons, and owls, adapted for ripping flesh Harvey Episodes, Videos, Games Pictures Nick Welcome OFFICIAL Harvey site with free online videos, clips, pics Come join Harvey, Fee Foo on some wild adventures Bustin Guide Service Bustin Hunting Service Powered by Optimization Company OurBizSpace Spring Snow Goose in Missouri Duck duck mrscienceut Bird Purpose In this activity, you will get a chance find out how shape s beak helps decide what it can eat Pretend Project Beak Adaptations Beaks Birds don t have teeth, paw, hands, antlers, horns, spines, but they do The also known as bill has two parts upper mandible lower World Of Owls Beaks, Feathers Flight Flight All short, downward facing that is hooked at end It designed specifically gripping tearing prey Toys Busy LLC ABC Acrylic Aronico Bagels Bogglers Balsa Bites Beads Kabobs Bizzy Birdy Playthings Cage n Queen Coconut Cracking Cracking usually thicker stronger than other size help indicate kind seed nut Mark Disney Wiki FANDOM powered Wikia Mark one reccurring antagonists DuckTales reboot He founder CEO Waddle, tech company located neighborhood Silverbeak industry billionaire, does not care much about money he cares his status, being buzzworthy, Debeaking Wikipedia Debeaking partial removal poultry, especially layer hens turkeys although may be performed quail ducksMost commonly, shortened permanently, regrowth occur trimmed somewhat longer Digital Services story unlikely friendship between kid who never broken rules friends ve lived any Bad Text Scott E McDonald BAD BEAKS PICTORIAL Key Point No matter deformed overgrown appears, Located Heart Mississippi Flyway, expect experience best spring snow goose hunting Just miles Southeast Poplar Bluff, Missouri, we offer exceptional hunts, doves hunts BEAKS High Frequency Words Flashcards (Collins Easy Learning KS1)Sneed B Collard III Official Site For another thing, Fall see release brand new Sneed books Warbers Woodpeckers available pre order from right now first my ever book adults, Warblers Father Son Big Year Birding Missoula author read days agoAward winning ventures into birding memoir book, Most Fun Book Ever About Lizards FREE shipping qualifying offers cool Literally They ectotherms, which means make their own heat That why many lizards basking sun Home Facebook III, Missoula, MT likes survived ridiculously long name become today top science Author Double Eagle biologist, world traveler, speaker, eighty young people, includ Third award along countless magazine articles both children adults marine biologist scientist training, most focus natural history, science, environment Beaks Collard, Robin Brickman , Paperback Young naturalists explore variety birds, habitats, them build, eat, survive From twisted crossbill color changing seagull, readers learn fun facts used tools birds all shapes sizes Teaching Nonfiction Revision Vicki Spandel blow roof off everything thought knew teaching nonfiction writing purposes revision Flash III include novels Dog Sense, Eagle, Governor Missing, Hangman Gold Beaks!
- 32 pages
- Sneed B Collard III
- 08 February 2016 Sneed B Collard III
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# Often asked: What Is Conclusion In Math?
## How do you write a math conclusion?
The middle section is all of the mathematical working. The Conclusion summarises your report giving information about the problem that you had to solve, the mathematical processes used to solve the problem, and discussion on how you solved the problem.
## What does conclusion mean?
1a: a reasoned judgment: inference The obvious conclusion is that she was negligent. b: the necessary consequence of two or more propositions taken as premises especially: the inferred proposition of a syllogism.
## How do we write a conclusion?
Conclusion outline
1. Topic sentence. Fresh rephrasing of thesis statement.
2. Supporting sentences. Summarize or wrap up the main points in the body of the essay. Explain how ideas fit together.
3. Closing sentence. Final words. Connects back to the introduction. Provides a sense of closure.
## What is the converse in math?
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. Either way, the truth of the converse is generally independent from that of the original statement.
You might be interested: How To Find Sales Tax In Math?
## What is an example of conclusion?
Sentence #1: restate the thesis by making the same point with other words (paraphrase). ~ Example: Thesis: “Dogs are better pets than cats.” Paraphrased: “Dogs make the best pets in the world.”
## How do you write a conclusion for a school project?
How to Write a Conclusion?
1. Restate the main premise or the main objectives.
2. Write one or two general sentences which accurately summarises the main body/arguments which support the main premise/theme of the work.
## What is the role of a conclusion?
Writing a Conclusion. A conclusion is an important part of the paper; it provides closure for the reader while reminding the reader of the contents and importance of the paper. A conclusion does not introduce new ideas; instead, it should clarify the intent and importance of the paper.
## What is another name for conclusion?
What is another word for conclusion?
end close
ending finish
cessation closure
finale halt
culmination denouement
## What is the root of conclusion?
Answer: The root of the word ” conclusion ” is “concl” which comes from the Latin “concludere”. Explanation: ” Conclusion ” originated from the Latin “concludere” and means act, process or effect of termination; finalization, termination. Conclusion is the act of finalizing an activity.
## How do you start a conclusion without conclusion?
Here are the 15 best alternatives ‘in conclusion ‘ to begin /transition to your conclusion:
1. In summary…
2. To sum up…
3. On the whole…
4. Overall, it may be said…
5. To conclude …
6. All things considered…
7. Finally…
8. Taking everything into account…
## What does a good conclusion look like?
A conclusion is, in some ways, like your introduction. You restate your thesis and summarize your main points of evidence for the reader. You can usually do this in one paragraph. In the following example, the thesis statement is in bold.
You might be interested: Often asked: What Is Function In Math?
## What words can I use to start a conclusion?
Transitional expressions
LOGICAL RELATIONSHIP TRANSITIONAL EXPRESSION
Conclusion /Summary finally, in a word, in brief, briefly, in conclusion, in the end, in the final analysis, on the whole, thus, to conclude, to summarize, in sum, to sum up, in summary
## Can the converse be true?
If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true. Example 1:
Statement If two angles are congruent, then they have the same measure.
Converse If two angles have the same measure, then they are congruent.
## What is a Contrapositive example?
Mathwords: Contrapositive. Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of “If it is raining then the grass is wet” is “If the grass is not wet then it is not raining.”
## What is the converse of the Pythagorean Theorem?
The converse of the Pythagorean Theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
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Termites are horrible pests, most of them blind and not seasonal, that infest a home and eat away at items made from wood or cellulose. They consume dead plant material and cellulose, such as wood, leaf matter, soil, and animal waste. Of the thousands of species of termites, five common types of termites exist such as drywood, conehead, damp wood, subterranean, and Formosan. The most common type that people identify with is drywood termites which feed on wood products.
Termites destroy soil structures and add to the problem of erosion. Termites serve a purpose, surviving in a variety of climates, such as subtropical and tropical regions. They recycle wood and plant matter which is important to the ecological cycle. They can colonize on most land areas except Antarctica. Termites, like cockroaches around since the beginning of human life as we know it, were once classified separate from cockroaches. However, recent studies indicate that termites come from close ancestors of cockroaches evolving during the Jurassic or Triassic periods millions of years ago. Existing are 3,106 species of termites. Similar to bees, wasps, and ants, termites divide and conquer their work by using castes that consist of sterile female and male “workers” or “soldiers”. Termite colonies range in size from a few hundred to millions. Now, that’s a scary thought.
Several predators including 65 species of birds and 19 types of mammals seek termites. They are hunted by arthropods like ants, centipedes, cockroaches (distant relatives), crickets, dragonflies, scorpions, spiders, reptiles (lizards), and amphibians (frogs and toads). Also, two species of spiders, from the family Ammoxenidae, find termites to be a tasty treat. The warning signs of a termite infestation include the following: when struck, solid wood sounds hollow, mud trails found on or nearby to exterior walls, and visible dry bubbled paint on furniture are all evident warning signs of a termite infestation. Termites feed upon dead plant materials, wood fixtures containing cellulose (sap) which termites find to be a delicious meal.
Use a professional exterminator service to destroy termites. Here is a list of home remedies toxic for termites: spray soapy water or use orange oil. Wet cardboard traps them and once trapped they can be sprayed with an insecticide or soapy water. Try using clove oil spray or a canola oil and water spray mixture. Garlic oil effectively destroys a large infestation. Mix garlic oil with water and spray an infested area with it or mix garlic oil with neem oil and tobacco. Aloe Vera gel and boric acid, non-toxic to humans, quickly kills termites. Petroleum jelly applied to furniture drives them away. Spray a mixture of white vinegar and lemon on a small infested area. Use a spray solution of salt and water. An age-old technique used to get rid of termites is to use sunlight exposure. Termites are averse to high temperature and extreme heat. Avoid using wood mulch for landscaping and plants. Remove old stumps and dead wood from your yard and avoid using it for landscaping purposes.
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Using tiny rust-containing spheres to tag cells, scientists from Johns Hopkins and elsewhere have successfully used magnetic resonance imaging to track stem cells implanted into a living animal, believed to be a first.
In the December issue of the journal Nature Biotechnology, the team said the neuronal stem cells take up and hold onto the spheres, which contain a compound of iron and oxygen. The iron-laden cells create a magnetic black hole easily spotted by magnetic resonance imaging, or MRI, they report.
"Until now, tissue had to be removed from an animal to see where stem cells were going, so this gives us an important tool," says author Jeff Bulte, Ph.D., associate professor of radiology at the Johns Hopkins School of Medicine. "Tracking stem cells non-invasively will likely be required as research advances, although human studies are still some time away."
Scientists at the University of Wisconsin School of Veterinary Medicine mixed the magnetic spheres, made by Trevor Douglas at Temple University, with stem cells that make the white matter, or neuronal covering, of the brain. Then they injected the iron-laden cells into the brains of rats that lack that covering. Using MRI scanners at the National Institutes of Health, Bulte watched the cells travel away from the injection site. The research was funded by the National Science Foundation, the Oscar Rennebohm Foundation and the Keck Foundation.
The rusty spheres, known as magnetic dendrimers, represent an important improvement over other magnetic tags, Bulte says. And even though the amount of iron used to label the cells is tiny compared to the total amount of iron in the body, the labeled cells stand out from other cells, magnetically speaking.
"During scanning, these labeled cells disturb the magnetic field created by the MRI machine, causing water molecules that pass by to get 'out of phase,'" he explains. "When this happens, the imaging scanner loses the signal, and the area looks black on the image."
Other researchers have used dendrimers containing gadolinium, which is also useful as a contrast medium for MRI, but which is toxic if it stays in the body for a prolonged time. But animal cells have a process to deal with iron and a storage mechanism for the metal, making the iron-based dendrimers inherently safer, says Bulte. For instance, iron is a key part of the transporter for oxygen and carbon monoxide found in red blood cells.
He adds that while it was not easy to develop the way to make magnetic dendrimers, it is easy to label cells with them. In essence, the dendrimer and the cell do that work themselves. Dendrimers stick to cells because they are charged -- kind of like static electricity. Cells then suck them inside and lock them away in the cellular equivalent of a garbage can -- a tiny holding spot called an endosome.
Other magnetic tags have used antibodies or other molecules that recognize and bind to certain features on cells, says Bulte. Unlike those tags, the magnetic dendrimers are universal; the scientists showed that different cell types will take in dendrimers just by mixing the spheres and the cells together, without affecting the cells' behavior.
Bulte's research with magnetic dendrimers is aligned with the Johns Hopkins Institute of Cell Engineering, created in early 2001 to advance research into the biology and potential application of pluripotent stem cells (primitive cells that become any type of cell in the body) and multipotent or adult stem cells (precursor cells that are naturally limited to becoming a specific tissue's cell types).
A next step with magnetic dendrimers, Bulte says, is watching the cells' distribution when they are injected into the circulatory system instead of the brain. Bulte also wants to study white blood cells in diseases of the central nervous system, such as multiple sclerosis, as well as the behavior of embryonic stem cells and stem cells from bone marrow. (Stem cells from bone marrow and blood have been used for decades in cancer treatments and more recently for some inherited metabolic disorders.)
Other co-authors of the report are Ian Duncan, Brian Witwer and Su-Chun Zhang of the University of Wisconsin School of Veterinary Medicine; Erica Strable of Temple University; Joseph Frank, Bobbi Lewis, Holly Zywicke, Brad Miller and Peter van Gelderen of the NIH; and Bruce Moskowitz of the Institute for Rock Magnetism at the University of Minnesota, Minneapolis. Bulte (NIH) and Douglas (Temple) have applied for a patent on these magnetic dendrimers.
On the Web: http://biotech.nature.com
Image available at: http://www.hopkinsmedicine.org/press/2001/DECEMBER/011214.htm
The above story is based on materials provided by Johns Hopkins Medical Institutions. Note: Materials may be edited for content and length.
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# Class 11 Maths NCERT Solutions for Chapter 3 Trigonometric Functions Exercise 3.1
### Trigonometric Functions Exercise 3.1 Solutions
1. Find the radian measures corresponding to the following degree measures:
(i) 25°
(ii) –47° 30'
(iii) 240°
(iv) 520°
Solution
(i) 25°
We know that 180° = Ï€ radian
(ii) -47° 30'
- 47° 30' = -47.5 degree [1° = 60']
= -95/2 degree
(iii) 240°
We know that 180° = Ï€ radian
(iv) 520°
We know that 180° = Ï€ radian
2. Find the degree measures corresponding to the following radian measures (use π = 22/7).
(i) 11/16
(ii) -4
(iii) 5Ï€/3
(iv) 7Ï€/6
Solution
(i) 11/16
We know that Ï€ radian = 180°
(ii) -4
We know that Ï€ radian = 180°
(iii) 5Ï€/3
We know that Ï€ radian = 180°
∴ 5Ï€/3 radian = 180/Ï€ × 5Ï€/3 degree = 300°
(iv) 7Ï€/6
We know that Ï€ radian = 180°
∴ 7Ï€/6 radian = 180/Ï€ × 7Ï€/6 = 210°
3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
Solution
Number of revolutions made by the wheel in 1 minute = 360
∴ Number of revolutions made by the wheel in 1 second = 360/60 = 6
In one complete revolution, the wheel turns an angle of 2Ï€ radian.
Hence, in 6 complete revolutions, it will turn an angle of 6 × 2Ï€ radian, i.e.,
Thus, in one second, the wheel turns an angle of 12Ï€ radian.
4. Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (use π = 22/7) .
Solution
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then
θ = 1/r
Therefore, for = 100 cm, l = 22 cm, we have
Thus, the required angle is 12° 36'.
5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
Solution
Diameter of the circle = 40 cm
∴ Radius (r) of the circle = 40/2 cm = 20 cm
Let AB be a chord (length = 20 cm) of the circle.
In ΔOAB, OA = OB = Radius of circle = 20 cm
Also, AB = 20 cm
Thus, ΔOAB is an equilateral triangle.
∴θ = 60° = Ï€/3 radian
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then θ = l/r.
Thus, the length of the minor arc of the chord is 20Ï€/3 cm.
6. If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
Solution
Let the radii of the two circles be and . Let an arc of length l subtend an angle of 60° at the centre of the circle of radius r1, while let an arc of length subtend an angle of 75° at the centre of the circle of radius r2.
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then θ = 1/r or l = rθ .
Thus, the ratio of the radii is 5 : 4.
7. Find the angle in radian though which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm
(ii) 15 cm
(iii) 21 cm
Solution
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then θ = l/r.
It is given that r = 75 cm
(i) Here, l = 10 cm
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1. Simple probability question. Involves drawing name for gift.
We had a dinner with my family where we draw names to give gift this christmas. The only rule was to not pick our name.
We did it a couple of time because we kept picking our name.
Finally we succeded and i started to think about what was the probability of someone picking his own name thus making this fail.
I built an excel simulation program which simulate drawing and got a consistent 50 % failure chance for 2 persons, about 66 % for 3 persons...and so on...but it's like there is a somekind of a maximum of about 68 % as i add people to my simulation.
For 6 persons i get 68 % failure and for 100 persons, it's about the same. I don't understand why that is.
So i started thinking...and the figures that i think it should be is about :
success rate should be 1/number of person, so 1/6 for 6 persons so a failure rate of 5/6....
Can anyone explain me what i should be getting ?
2. Originally Posted by moris7
We had a dinner with my family where we draw names to give gift this christmas. The only rule was to not pick our name
We did it a couple of time because we kept picking our name.
Finally we succeded and i started to think about what was the probability of someone picking his own name thus making this fail.
Just to be clear on the question: “Six people draw names from the hat.
What is the probability that no one draws his/her own name?
If that is the question, there is a well-known answer. These are derangements.
The number of derangements on six objects is $D(6)=265$.
So the probability no one has a match is $\frac{265}{6!}=0.3681$.
Subtracting that number from 1, gives the probability of at least one match.
BTW: If $n\ge 3$ then a very close approximation of $D(n) \approx \frac{{n!}}{e}$.
$\frac{6!}{e}=264.873197643438$
3. Thank you very much for your answer...this seems like the right answer...i will try to see why my simulation does not point exactly toward this number.
4. It's because of excel random function which is clearly not random.
5. a descriptive title
I solved it once without derangements... think of this name-drawing as a permutation where each spot represents a person... and then those spots get filled in with their names (ie 1 through 6) in some order.... 123456 143265 654312 etc... being in the correct spot represents drawing your own name.
• Number of permutations where all 6 draw their name: 1
• Number of permutations where exactly five draw their own name: 0 (it's impossible)
• Number of permutations where exactly four draw their own name: 6C4*1 = 15
• Number of permutations where exactly three draw their own name: 6C3*2=40 (6C3 ways to choose which three draw their own names and 2 ways for the remaining people to not draw their names)
• Number of permutations where exactly two draw their own name: 6C2*Q where Q=the number of ways for four people to not draw their own name.
To get Q you have to recursively use the previous steps... think permutations like 1243 1234 etc.
permutations (length 4) where all 4 draw their own name: 1
permutations where exactly 3 draw their own name: 0
permutations where exactly 2 draw their own name: 4C2*1=6
permutations where exactly 1 draws their own name: 4C1*2=8
permutations where none draw their own name: (4x3x2x1)-(1+6+8)=9
therefore Q=9
6C2*Q=6C2*9= 135 and therefore there are 135 permutations (length 6) where exactly 2 people draw their own name.
• number of permutations where exactly 1 person draws their own name: 6C1*L where L is the number of ways for 5 people to not draw their own names
permutations (length 5) where all 5 draw their own name: 1
permutations where exactly 4 draw their own name: 0
permutations where exactly 3 draw their own name: 5C3*1= 10
permutations where exactly 2 draw their own name: 5C2*2= 20
permutations where exactly 1 draws their own name: 5C1*Q=5C1*9= 45
permutations where none draw their own name: (5x4x3x2x1)-(1+10+20+45)=44
therefore L=44
6C1*L=6C1*44= 264 and therefore there are 264 permutations (length 6) where exactly 1 person draws their own name.
Therefore, the probability of no one drawing their own name in a group of six people is:
1-P(1 or 2 or 3 or 4 or 5 or 6 people draw their own name)
=1- (1+15+40+135+264)/(6x5x4x3x2x1)
=0.36805555555
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Robots are performing complex quality control tasks at waste management facilities that are usually done by humans, with surprising results
This article first appeared on Business Insider
- Only a small fraction of the garbage the world produces each year gets recycled – about 16% – and that number has gotten even smaller in the past year.
- About one-fourth of items that are put in recycling bins can’t be recycled at all, including greasy pizza boxes.
- Recycling takes up so much resources to sort and process, US cities resort to burning or trashing recyclable items to save money.
- Artificial intelligence companies are enlisting robots to make recycling sustainable, by sorting through trash themselves.
Humans have enlisted nearly 100 AI-powered robots in North American to come to the rescue for something humans are terrible at: recycling.
Even when we try to do it right, we’re often making things worse; About one out of every four of the things people throw into the recycling bin aren’t recyclable at all.
All those misplaced greasy pizza boxes (not recyclable) and clamshell containers tossed in with the plastics, have imperiled an industry that was never really that effective in the first place.
Only a small fraction of the over 2.1 billion tons of the garbage the world produces each year gets recycled – about 16%.
And even that small sliver has gotten smaller over the past year.
Since China no longer processes other countries’ waste, other countries need to find an alternative to burning its trash
For decades, the US sold more than half of its recyclables to China – mostly plastics to be melted into pellets, the raw material for making more plastic.
But in March of 2018, China said, “No More.”
“They started shipping more and more stuff to China, often contaminated dirty plastics or mixed too many mixed goods,” said Kate O’Neill, a UC Berkeley professor and author of “Waste.”
Around a quarter of the shipments China received had to be hand-processed, buried in landfills, or incinerated.
So the Chinese government declared that bales could contain only up to half a percent of things that contaminated them, like food wrappers or a dirty jar of peanut butter. US consumers and recycling centers couldn’t keep up.
“I think people in the wealthy countries had gotten complacent, never bothering to build more recycling facilities domestically,” O’Neill added.
Installing artificial intelligence in robots to sort through trash may be a sustainable solution
Today, a handful of start-ups are testing out new technology to make recycling sustainable.
AMP Robotics is an artificial intelligence and robotics company that aims to change the way we recycle.
Founder of AMP Robotics, Matanya Horowitz said “the situation with the Chinese export markets have actually been good for [the company].”
AMP Robotics is rolling out its latest model: a “Cortex Robot” that uses optical sensors to take in what rolls by, and a “brain” to figure out what his “hands” should do with something – even if it looks different to anything he’s seen before.
“A lot of these recycling facilities are structured with the primary task of basically dealing with contamination that’s not supposed to be there,” said Horotwiz. “”What we see is a lot of recycling facilities are investing in automation to help improve their operations.”
At least four companies are rolling out similar models, in the hopes of turning a profit from the US’ growing piles of hard-to-sort recyclables.
And investors are taking notice. In November 2019, AMP Robotics announced a $16 million Series A investment from Sequoia Capital.
China is helping its own citizens get better at recycling
But what about helping humans get better at choosing what to put in their recycling bins in the first place?
New policies in Shanghai are one of the first steps in China’s push to solve its waste problems.
This past summer, citizens will face fines and what are called “social penalties” if they don’t sort things properly.
One trash sorting volunteer said, Shanghai started the test run on June 24. “It was very hard for us at the beginning. Everyone was busy, people didn’t know how to sort,” the volunteer – who requested to be unidentified – said.
“At first we had some hard times,” said Shanghai citizen Zhaoju Zhang. “The most difficult part was how to differentiate between dry and wet trash. It was so complicated that we all got confused.”
Almost immediately, hundreds of AI-enabled apps sprouted up in order to assist everyday sorting.
“If it’s something that is confusing whether it’s dry or wet trash, we can just scan the item and get the answer,” Zhang said.
But not everyone has access to AI to help parse the new rules, and many complain that complying is tough, and punishments are too harsh.
Kate O’Neill said the new laws are having a “massive cultural impact” and there are “some concerns about how draconian it is, but it’s too early to really tell the results. But it certainly has seems to be a massive culture shift.”
This kind of cultural shift in how we throw things away would be challenging in the US, where the average person produces twice as much trash as a Chinese citizen.
But experts warn that rethinking the way we deal with garbage is essential, and AI technology offers a promising way forward.
It’s even possible for it to identify who created a piece of trash in the first place.
Horowitz explained that robots are able to learn the features of materials. They are able to sparse whether a material is cloudy or opaque. AI robots may even be able to identify symbols of specific brands. All of these abilities help the robots like Max narrow down the source of contamination and what to do with it.
Last year, over 250 companies signed a MacArthur Foundation agreement pledging that 100% of plastic packaging will be easily and safely reused, recycled, or composted by 2025.
CEO of SC Johnson, Fisk Johnson, said in an interview, “We’re a family company, and we have a very long-term view, and business has to be part of the solution.”
Whether or not they make good on this pledge, AI will be quietly watching, and gathering data on the packaging these brands continue to use.
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# Optics - distance between object and image
1. Jul 21, 2009
### Smile101
hello,
1. The problem statement, all variables and given/known data
A lens has a focal length of 20 cm and a magnification of 4. How far apart are the object and the image?
2. Relevant equations
m=di(image)/do(object)
1/di + 1/do = 1/f
3. The attempt at a solution
Well, I don't really know what to do, I feel like I don't have enough information. Your help will be very appreciated!
2. Jul 21, 2009
### Redbelly98
Staff Emeritus
Re: Optics-Lens
Since you have 2 equations in 2 unknowns ... solve 1 equation for 1 of the unknowns, then substitute that into the 2nd equation.
3. Jul 21, 2009
### Smile101
Re: Optics-Lens
Yes, I was thinking about that but how do I do it? Seeing as if i did m=di/do 4=di/do --but it leads me to a dead end
or if i used 1/di+1/do=1/f i can only go as far as 1/di+1/do=1/20 then once again i get to a dead end!
4. Jul 21, 2009
### Redbelly98
Staff Emeritus
Re: Optics-Lens
Okay, so far so good. Now solve that equation for di:
di = _____ ?
5. Jul 21, 2009
### Smile101
Re: Optics-Lens
umm.. di=4(do) but since we dont have do we can't solve for di
6. Jul 21, 2009
### Redbelly98
Staff Emeritus
Re: Optics-Lens
You can now substitute 4do for di in the other equation ... that eliminates di, leaving do as the only unknown.
7. Jul 21, 2009
### Smile101
Re: Optics-Lens
so the equation is : 1/di +1/do =1/f
1/4do + 1/do = 1/20
Now I'm guessing we want to make do single... but i have to get rid of the 4 so since i'm dividing....
1/do^2 = 1/20 - 1/4
8. Jul 21, 2009
### rl.bhat
Re: Optics-Lens
Multiply do to both sides and solve for do.
9. Jul 22, 2009
### Redbelly98
Staff Emeritus
Re: Optics-Lens
Yes, exactly. It's just algebra at this point.
10. Jul 25, 2009
### Smile101
Re: Optics-Lens
sorry for the late reply but, thanks a lot for your help, I really appreciated it! :)
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## Archive for the 'Bad Algebra' category
And indeed, he was right. Phil Plait the Bad Astronomer, of all people, got taken in by a bit of mathematical stupidity, which he credulously swallowed and chose to stupidly expand on.
We'll consider three infinite series:
S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...
S2 = 1 - 2 + 3 - 4 + 5 - 6 + ...
S3 = 1 + 2 + 3 + 4 + 5 + 6 + ...
S1 is something called Grandi's series. According to the video, taken to infinity, Grandi's series alternates between 0 and 1. So to get a value for the full series, you can just take the average - so we'll say that S1 = 1/2. (Note, I'm not explaining the errors here - just repeating their argument.)
Now, consider S2. We're going to add S2 to itself. When we write it, we'll do a bit of offset:
1 - 2 + 3 - 4 + 5 - 6 + ...
1 - 2 + 3 - 4 + 5 + ...
==============================
1 - 1 + 1 - 1 + 1 - 1 + ...
So 2S2 = S1; therefore S2 = S1=2 = 1/4.
Now, let's look at what happens if we take the S3, and subtract S2 from it:
1 + 2 + 3 + 4 + 5 + 6 + ...
- [1 - 2 + 3 - 4 + 5 - 6 + ...]
================================
0 + 4 + 0 + 8 + 0 + 12 + ... == 4(1 + 2 + 3 + ...)
So, S3 - S2 = 4S3, and therefore 3S3 = -S2, and S3=-1/12.
So what's wrong here?
To begin with, S1 does not equal 1/2. S1 is a non-converging series. It doesn't converge to 1/2; it doesn't converge to anything. This isn't up for debate: it doesn't converge!
In the 19th century, a mathematician named Ernesto Cesaro came up with a way of assigning a value to this series. The assigned value is called the Cesaro summation or Cesaro sum of the series. The sum is defined as follows:
Let . In this series, . is called the kth partial sum of A.
The series is Cesaro summable if the average of its partial sums converges towards a value .
So - if you take the first 2 values of , and average them; and then the first three and average them, and the first 4 and average them, and so on - and that series converges towards a specific value, then the series is Cesaro summable.
Look at Grandi's series. It produces the partial sum averages of 1, 1/2, 2/3, 2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, ... That series clearly converges towards 1/2. So Grandi's series is Cesaro summable, and its Cesaro sum value is 1/2.
The important thing to note here is that we are not saying that the Cesaro sum is equal to the series. We're saying that there's a way of assigning a measure to the series.
And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn't true.
From there, they go on to start playing with the infinite series in sloppy algebraic ways, and using the Cesaro summation value in their infinite series algebra. This is, similarly, not a valid thing to do.
Just pull out that definition of the Cesaro summation from before, and look at the series of natural numbers. The partial sums for the natural numbers are 1, 3, 6, 10, 15, 21, ... Their averages are 1, 4/2, 10/3, 20/4, 35/5, 56/6, = 1, 2, 3 1/3, 5, 7, 9 1/3, ... That's not a converging series, which means that the series of natural numbers does not have a Cesaro sum.
What does that mean? It means that if we substitute the Cesaro sum for a series using equality, we get inconsistent results: we get one line of reasoning in which a the series of natural numbers has a Cesaro sum; a second line of reasoning in which the series of natural numbers does not have a Cesaro sum. If we assert that the Cesaro sum of a series is equal to the series, we've destroyed the consistency of our mathematical system.
Inconsistency is death in mathematics: any time you allow inconsistencies in a mathematical system, you get garbage: any statement becomes mathematically provable. Using the equality of an infinite series with its Cesaro sum, I can prove that 0=1, that the square root of 2 is a natural number, or that the moon is made of green cheese.
What makes this worse is that it's obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn't matter that infinity is involved: you can't following a monotonically increasing trend, and wind up with something smaller than your starting point.
Someone as allegedly intelligent and educated as Phil Plait should know that.
## Genius Continuum Crackpottery
There's a lot of mathematical crackpottery out there. Most of it is just pointless and dull. People making the same stupid mistakes over and over again, like the endless repetitions of the same-old supposed refutations of Cantor's diagonalization.
After you eliminate that, you get reams of insanity - stuff which
is simply so incoherent that it doesn't make any sense. This kind of thing is usually word salad - words strung together in ways that don't make sense.
After you eliminate that, sometimes, if you're really lucky, you'll come accross something truly special. Crackpottery as utter genius. Not genius in a good way, like they're an outsider genius who discovered something amazing, but genius in the worst possible way, where someone has created something so bizarre, so overwrought, so utterly ridiculous that it's a masterpiece of insane, delusional foolishness.
Today, we have an example of that: Existics!. This is a body of work by a high school dropout named Gavin Wince with truly immense delusions of grandeur. Pomposity on a truly epic scale!
I'll walk you through just a tiny sample of Mr. Wince's genius. You can go look at his site to get more, and develop a true appreciation for this. He doesn't limit himself to mere mathematics: math, physics, biology, cosmology - you name it, Mr. Wince has mastered it and written about it!
The best of his mathematical crackpottery is something called C3: the Canonized Cardinal Continuum. Mr. Wince has created an algebraic solution to the continuum hypothesis, and along the way, has revolutionized number theory, algebra, calculus, real analysis, and god only knows what else!
Since Mr. Wince believes that he has solved the continuum hypothesis. Let me remind you of what that is:
1. If you use Cantor's set theory to explore numbers, you get to the uncomfortable result that there are different sizes of infinity.
2. The smallest infinite cardinal number is called ℵ0,
and it's the size of the set of natural numbers.
3. There are cardinal numbers larger than ℵ0. The first
one larger than ℵ0 is ℵ1.
4. We know that the set of real numbers is the size of the powerset
of the natural numbers - 20 - is larger than the set of the naturals.
5. The question that the continuum hypothesis tries to answer is: is the size
of the set of real numbers equal to ℵ1? That is, is there
a cardinal number between ℵ0 and |20|?
The continuum hypothesis was "solved" in 1963. In 1940, Gödel showed that you couldn't disprove the continuum hypothesis using ZFC. In 1963,
another mathematician named Paul Cohen, showed that it couldn't be proven using ZFC. So - a hypothesis which is about set theory can be neither proven nor disproven using set theory. It's independent of the axioms of set theory. You can choose to take the continuum hypothesis as an axiom, or you can choose to take the negation of the continuum hypothesis as an axiom: either choice is consistent and valid!
It's not a happy solution. But it's solved in the sense that we've got a solid proof that you can't prove it's true, and another solid proof that you can't prove it's false. That means that given ZFC set theory as a basis, there is no proof either way that doesn't set it as an axiom.
But... Mr. Wince knows better.
The set of errors that Wince makes is really astonishing. This is really seriously epic crackpottery.
He makes it through one page without saying anything egregious. But then he makes up for it on page 2, by making multiple errors.
First, he pulls an Escultura:
x1 = 1/21 = 1/2 = 0.5
x2 = 1/21 + 1/22 = 1/2 + 1/4 = 0.75
x3 = 1/21 + 1/22 + 1/23 = 1/2 + 1/4 + 1/8 = 0.875
...
At the end or limit of the infinite sequence, the final term of the sequence is 1.0
...
In this example we can see that as the number of finite sums of the sequence approaches the limit infinity, the last term of the sequence equals one.
xn = 1.0
If we are going to assume that the last term of the sequence equals one, it can be deduced that, prior to the last term in the sequence, some finite sum in the series occurs where:
xn-1 = 0.999…
xn-1 = 1/21 + 1/22 + 1/23 + 1/24 + … + 1/2n-1 = 0.999…
Therefore, at the limit, the last term of the series of the last term of the sequence would be the term, which, when added to the sum 0.999… equals 1.0.
There is no such thing as the last term of an infinite sequence. Even if there were, the number 0.999.... is exactly the same as 1. It's a notational artifact, not a distinct number.
But this is the least of his errors. For example, the first paragraph on the next page:
The set of all countable numbers, or natural numbers, is a subset of the continuum. Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.
We didn't need to go through the difficult of Cantor's diagonalization! We could have just blindly asserted that it's obvious!
or actually... The fact that there are multiple degrees of infinity is anything but obvious. I don't know anyone who wasn't surprised the first time they saw Cantor's proof. It's a really strange idea that there's something bigger than infinity.
Moving on... the real heart of his stuff is built around some extremely strange notions about infinite and infinitessimal values.
Before we even look at what he says, there's an important error here
which is worth mentioning. What Mr. Wince is trying to do is talk about the
continuum hypothesis. The continuum hypothesis is a question about the cardinality of the set of real numbers and the set of natural numbers.
Neither infinites nor infinitessimals are part of either set.
Infinite values come into play in Cantor's work: the cardinality of the natural numbers and the cardinality of the reals are clearly infinite cardinal numbers. But ℵ0, the smallest infinite cardinal, is not a member of either set.
Infinitessimals are fascinating. You can reconstruct differential and integral calculus without using limits by building in terms of infinitessimals. There's some great stuff in surreal numbers playing with infinitessimals. But infinitessimals are not real numbers. You can't reason about them as if they were members of the set of real numbers, because they aren't.
Many of his mistakes are based on this idea.
For example, he's got a very strange idea that infinites and infinitessimals don't have fixed values, but that their values cover a range. The way that he gets to that idea is by asserting the existence
of infinity as a specific, numeric value, and then using it in algebraic manipulations, like taking the "infinityth root" of a real number.
For example, on his way to "proving" that infinitessimals have this range property that he calls "perambulation", he defines a value that he calls κ:
In terms of the theory of numbers, this is nonsense. There is no such thing as an infinityth root. You can define an Nth root, where N is a real number, just like you can define an Nth power - exponents and roots are mirror images of the same concept. But roots and exponents aren't defined for infinity, because infinity isn't a number. There is no infinityth root.
You could, if you really wanted to, come up with a definition of exponents that that allowed you to define an infinityth root. But it wouldn't be very interesting. If you followed the usual pattern for these things, it would be a limit: . That's clearly 1. Not 1 plus something: just exactly 1.
But Mr. Cringe doesn't let himself be limited by silly notions of consistency. No, he defines things his own way, and runs with it. As a result, he gets a notion that he calls perambulation. How?
Take the definition of κ:
Now, you can, obviously, raise both sides to the power of infinity:
Now, you can substitute ℵ0 for . (Why? Don't ask why. You just can.) Then you can factor it. His factoring makes no rational sense, so I won't even try to explain it. But he concludes that:
• Factored and simplified one way, you end up with (κ+1) = 1 + x, where x is some infinitessimal number larger than κ. (Why? Why the heck not?)
• Factored and simplified another way, you end up with (κ+1) = ℵ
• If you take the mean of of all of the possible factorings and reductions, you get a third result, that (κ+1) = 2.
He goes on, and on, and on like this. From perambulation to perambulating reciprocals, to subambulation, to ambulation. Then un-ordinals, un-sets... this is really an absolute masterwork of utter insane crackpottery.
• Scientopia Blogs
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# Need help on integration question i found on net
1. Sep 26, 2009
### Keval
Question is in orange, answer is in black.
I got no idea how they got this answer :\
The way im trying is using a substition of $kx^2 = gsin\theta$
Just the one question i cant get my head aroung in this exercise step by step method would be appreciated
2. Sep 26, 2009
### tiny-tim
Hi Keval!
Not sin, but tanh … try substituting x√(k/g) = tanhu
3. Sep 26, 2009
### g_edgar
With denominator instead k x^2 + g you get an arctan answer, right? Use the same steps on this one and get an atanh answer.
4. Sep 26, 2009
### Keval
someone check this out for me ??
$-\frac{1}{k} \int \frac{dx}{x^2-\frac{g}{k}}$
to simplify i let$\frac{g}{k}=m^2$ to get
$-\frac{1}{k}\int \frac{dx}{x^2-m^2}=-\frac{1}{k}\int \frac{dx}{(x-m)(x+m)}$
Using partial fractions i got
$\int \frac{dx}{x^2-m^2}=\int \frac{\frac{1}{2m}}{x-m}dx+\frac{-\frac{1}{2m}}{x+m}dx=$
$\frac{1}{2m}\ln|x-m|-\frac{1}{2m}\ln|x+m|=\frac{1}{2m}\ln \left| \frac{x-m}{x+m}\right|=$
$-\frac{1}{m}\cdot \frac{1}{2}\ln \left|\frac{x+m}{x-m} \right|=-\frac{1}{m}\tanh^{-1}\left( \frac{x}{m}\right)$
$\frac{1}{km}\tanh^{-1}\left( \frac{x}{m}\right)=\frac{1}{\sqrt{gk}}\tanh^{-1}\left(\sqrt{\frac{k}{g}}x \right)$
5. Sep 27, 2009
### tiny-tim
Hi Keval!
Yes, your partial fractions method is fine.
But it would be far quicker to start "let x√(k/g) = tanhu, dx√(k/g) = sech2u du" …
try it!
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crawl-data/CC-MAIN-2017-43/segments/1508187822992.27/warc/CC-MAIN-20171018142658-20171018162658-00834.warc.gz
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# Root-Mean-Square Value of Voltage and Current
Applying a sinusoidal voltage u(t) to a resistive load R causes the following current to flow through the load according to Ohm's law:
Because the voltage and current are time-dependent variables, so is the power produced in the resistor. It is defined by the following equation:
The diagrams below show the time characteristics of an AC voltage and current (upper diagram) along with the power (lower diagram).
The area enclosed between the power curve and time axis is a measure of the electrical energy converted by the resistor into heat. If a horizontal line is drawn parallel to the time axis at a height of p0/2, the areas above and below this line respectively (shaded in matching colours below) are equal in size. An average power p0/2 ascertained in this manner over several periods of oscillation would perform the same amount of work as the continuously changing instantaneous power p(t) does. This is illustrated by the diagram below.
The following animation demonstrates this relationship.
A DC voltage U that would be needed to develop the same power as the AC voltage in the resistor is determined as follows:
Resolving this equation in terms of the DC voltage U gives
This voltage U is termed the root mean square value of the alternating signal. Because it is a time-independent variable, it is designated in uppercase just like a direct voltage. Root mean square values of alternating current are specified in the same manner. In other words:
The rms values U and I of AC voltage and current in a resistor R develop the same power P as a DC current I and voltage U of equal magnitude.
The relationship between the rms and peak values of current and voltage in the case of the sinusoidal variables considered here is given by the following equations:
Accordingly, the rms value of a voltage or current is about 70% of the peak value.
Example: a mains voltage with an rms value U = 220 V has a peak value
Naturally, the rms values of non-sinusoidal periodic signals like triangular and rectangular forms can also be defined. In such cases, however, the mathematical relationship (i.e. conversion ratio) between the rms and peak (amplitude) values varies in accordance with the signal shape under consideration.
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crawl-data/CC-MAIN-2022-27/segments/1656104141372.60/warc/CC-MAIN-20220702131941-20220702161941-00776.warc.gz
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Ehlers-Danlos syndrome (EDS) is a group of hereditary connective tissue disorders characterized by defects of the major structural protein in the body (collagen). Collagen, a tough, fibrous protein, plays an essential role in “holding together,” strengthening, and providing elasticity to bodily cells and tissues. Due to defects of collagen, primary EDS symptoms and findings include abnormally flexible, loose joints (articular hypermobility) that may easily become dislocated; unusually loose, thin, “stretchy” (elastic) skin; and excessive fragility of the skin, blood vessels, and other bodily tissues and membranes.
The different types of EDS were originally categorized in a classification system that used Roman numerals (e.g., EDS I to EDS XI), based upon each form’s associated symptoms and findings (clinical evidence) and underlying cause. A revised, simplified classification system (revised nosology) has since been described in the medical literature that categorizes EDS into six major subtypes, based upon clinical evidence, underlying biochemical defects, and mode of inheritance.
Each subtype of EDS is a distinct hereditary disorder that may affect individuals within certain families (kindreds). In other words, parents with one subtype of EDS will not have children with another EDS subtype. Depending upon the specific subtype present, Ehlers-Danlos syndrome is usually transmitted as an autosomal dominant or autosomal recessive trait.
The symptoms and findings associated with Ehlers-Danlos syndrome (EDS) may vary greatly in range and severity from case to case, depending upon the specific form of the disorder present and other factors. However, the primary findings associated with EDS typically include abnormal “looseness” (laxity) and excessive extension (hyperextension) of joints; susceptibility to partial or complete joint dislocations; chronic joint pain; a tendency to develop degenerative joint disease (osteoarthritis) at an early age; unusually loose, thin, elastic skin; and excessive fragility of the skin, blood vessels, and other bodily tissues and membranes. Due to tissue fragility, affected individuals may easily bruise; experience prolonged bleeding (hemorrhaging) after trauma; have poor wound healing; develop “parchment-like,” thin scarring; and/or have other associated abnormalities.
In many individuals with EDS, associated symptoms and findings may become apparent during childhood. More rarely, depending upon the specific disorder subtype present, certain abnormalities may be apparent beginning at birth (congenital). In addition, in other individuals, such as those with mild disease manifestations, the disorder may not be recognized until adulthood.
The different forms of EDS were formally classified in the 1980s using a Roman numeral system. This categorization identified at least 10 major forms of the disorder based upon genetic and biochemical abnormalities as well as associated symptoms and findings. However, a simplified, revised, updated classification system has since been published in the medical literature that classifies EDS into six primary subtypes as well as some other forms of EDS, based upon the specific underlying biochemical cause, mode of inheritance, major and minor symptoms, and physical findings. The revised classification system serves to further differentiate between the various forms of the disorder as well as some related disorders.
The original classification system differentiates between severe and mild forms of classic EDS (EDS I and II). In the revised categorization, EDS I and II are reclassified as one subtype, known as EDS classical type. According to reports in the medical literature, in individuals with this subtype, associated skin abnormalities may vary greatly, ranging from mild, moderate, to severe in certain affected families (kindreds). EDS classical type may be characterized by excessive laxity and extension of the joints (hypermobility); susceptibility to recurrent sprains and dislocations of certain joints, such as the knees and shoulders; abnormally increased elasticity and extension (hyperextensibility) of the skin; and tissue fragility, potentially leading to degeneration or “splitting” of the skin, abnormal healing of skin wounds, and characteristic, thin, “parchment-” or “paper-like” (papyraceous) scarring that often becomes discolored and widened. Such scarring may occur primarily over certain prominent bony areas (pressure points), such as the shins, knees, elbows, and forehead. In individuals with EDS classical type, additional findings may include the formation of relatively small, fleshy, tumor-like skin growths (molluscoid pseudotumors) and/or hard, round, movable lumps (calcified spheroids) under the skin; unusually “velvety” skin; diminished muscle tone (hypotonia); and/or flat feet (pes planus). EDS classical type may also be characterized by easy bruisability, often occurring in the same areas; abnormal displacement (prolapse) of certain organs due to tissue fragility, such as protrusion of part of the stomach upward through an opening in the diaphragm (hiatal hernia); and/or an increased risk of certain complications after surgical procedures. For example, postsurgical complications may include protrusion of certain organs through weak areas in surrounding membranes, muscles, or other tissues (postsurgical hernias). In addition, some individuals with this subtype may have a deformity of one of the heart valves (mitral valve prolapse), allowing blood to leak backwards into the left upper chamber of the heart (mitral insufficiency), and/or, more rarely, abnormal widening (dilatation) of a region of the aorta, the major blood vessel of the body.
EDS hypermobility type was formerly classified as EDS III or benign hypermobility syndrome. This form of the disorder is primarily characterized by generalized, excessive extension (hypermobility) of the large and small joints. Additional findings may include abnormally increased skin elasticity, an unusually smooth or “velvet-like” consistency of the skin, and/or easy bruising. Skin abnormalities and bruising susceptibility may be extremely variable from case to case. Some individuals with EDS hypermobility type may develop chronic, potentially disabling joint pain and be prone to recurrent dislocations, particularly of the knee, shoulder, and jaw (i.e., temporomandibular) joints.
EDS vascular type (formerly EDS IV or EDS arterial-ecchymotic type) is primarily characterized by unusually thin, transparent skin with prominent underlying veins, particularly in the chest and abdominal areas; a susceptibility to severe bruising from minor trauma; and tissue fragility, potentially resulting in spontaneous rupture of certain membranes and tissues. For example, affected individuals may be prone to spontaneous rupture of certain mid-sized or large arteries or the intestine (bowel), leading to life-threatening complications. Because acute pain in the abdominal or flank area may indicate possible arterial or intestinal rupture, such symptoms require immediate, emergency medical attention. Individuals with EDS vascular type may also be prone to developing abnormal channels between certain arteries and veins (arteriovenous fistula, e.g., carotid-cavernous sinus fistula) and have an increased risk of weakening of arterial walls and associated bulging of certain arteries (aneurysms), such as those supplying the head and neck (carotid arteries) and within the skull (intracranial). Aneurysms may be prone to rupturing, potentially resulting in life-threatening complications. Females with EDS vascular type may also be at risk for arterial bleeding and rupture of the uterus during pregnancy as well as vaginal tearing, uterine rupture, and/or other complications during delivery. In addition, affected individuals may be prone to experiencing certain complications during and after surgical procedures, such as separation of the layers of a surgical wound (dehiscence).
Individuals with EDS vascular type may also have abnormally decreased levels of fatty tissue under skin layers (subcutaneous adipose tissue) of the hands, arms, legs, feet, and face. As a result, some affected individuals may have a characteristic facial appearance, including thin lips; a thin, pinched nose; relatively large, prominent eyes; hollow cheeks; and tight, lobeless ears. In addition, skin of the hands and feet may appear prematurely aged (acrogeria). Additional symptoms and findings associated with this EDS subtype may include a deformity in which the foot is twisted out of position at birth (clubfoot); hypermobility that may be limited to joints of the fingers and toes (digits); the early onset of varicose veins, which are unusually widened, twisted veins visible under the skin; and spontaneous rupture of muscles and tendons. In addition, some with this EDS subtype may be susceptible to abnormal accumulations of air and blood in the chest cavity (pneumohemothorax) and/or associated collapse of the lungs (pneumothorax).
In individuals with EDS kyphoscoliosis type (formerly EDS VI), certain symptoms and findings may be apparent at birth (congenital). These include abnormal sideways curvature of the spine (congenital scoliosis) that becomes progressively severe; diminished muscle tone (hypotonia); and generalized, excessive extension and looseness (laxity) of the joints. In children with the disorder, severe hypotonia may cause delays in the acquisition of certain motor skills, and affected adults may lose the ability to walk by the second or third decade of life. Additional findings associated with EDS kyphoscoliosis type may include easy bruising, tissue fragility and associated degenerative (atrophic) scarring of the skin, a risk of spontaneous arterial rupture, abnormally reduced bone mass (osteopenia), and unusually small corneas (microcornea). In addition, because the opaque, inelastic membrane covering the eyeballs (sclera) may be unusually fragile, minor trauma may result in rupture of the sclera, rupture of the transparent region in the front of the eyes (cornea), and/or detachment of the nerve-rich membrane in the back of the eyes (retina).
EDS arthrochalasia type (formerly EDS VII, Autosomal Dominant [EDS VIIA and VIIB]) is primarily characterized by dislocation of the hips at birth (congenital hip dislocation); severe, generalized, excessive extension of the joints (hypermobility); and recurrent partial dislocations of affected joints (subluxations), such as those of the elbows, knees, hips, and feet. Affected individuals may also have diminished muscle tone (hypotonia), abnormal front-to-back and sideways curvature of the spine (kyphoscoliosis), and mildly reduced bone mass (osteopenia). Additional findings typically include abnormally increased elasticity and extension of the skin (hyperextensibility), easy bruising, and tissue fragility, with associated scarring of the skin.
Primary symptoms and findings associated with EDS dermatosparaxis type (formerly EDS VII, Autosomal Recessive [EDS VIIC]) include severe skin fragility; soft, sagging, redundant skin; and extensive bruising. In some cases, certain tissues or organs may abnormally protrude through a weak area in a surrounding membrane, muscle, or other tissue (e.g., umbilical hernia, inguinal hernia).
In addition to the six primary EDS subtypes described above, there are some additional, rare forms of EDS. For example, X-linked EDS (formerly EDS Type V) has been described in individuals within at least one family (kindred). Associated symptoms and findings include easy bruising, hyperextensible skin, minor skin fragility, and deformity of one of the heart valves (mitral valve prolapse), allowing blood to leak backwards into the left upper chamber of the heart (mitral insufficiency). Because this form of EDS is transmitted as an X-linked recessive trait, it is fully expressed in males only. (For more information on X-linked inheritance, please see the “Causes” section of this report below.)
The symptoms and findings associated with EDS periodontosis type (formerly EDS Type VIII) are considered similar to those seen in EDS classical type. Additional findings typically include disease of the tissues surrounding and supporting the teeth (periodontal disease), potentially resulting in premature tooth loss.
EDS progeroid form, another rare variant of the disorder, is characterized by loose, elastic skin; hypermobile joints; slow wound healing; degenerative (atrophic) skin scars; and reduced bone mass (osteopenia). Additional findings may include delayed mental development, short stature, and a prematurely aged appearance (progeroid appearance) due to premature wrinkling of facial skin; scanty scalp hair, eyebrows, and eyelashes; and other findings.
According to reports in the literature, some individuals may be affected by additional, rare subtypes of EDS, which are currently referred to as EDS unspecified forms. Such subtypes are characterized by joint hypermobility, loose, elastic skin, and other symptoms and findings commonly seen in individuals with the disorder.
The EDS subtype originally referred to as EDS type X (or EDS dysfibronectinemic type) is extremely rare, affecting only one reported family (kindred). This subtype is characterized by abnormally extensible, loose joints; thin, elastic skin; and abnormalities of the specialized blood cells that play an essential role in blood clotting (platelets). Associated findings typically include the appearance of tiny purplish or reddish spots on the skin due to abnormal bleeding within or under skin layers (petechiae) and/or pinkish, depressed scar-like skin lesions that may later become white (striae distensae). These lesions, which may occur on the thighs, abdomen, buttocks, and breasts, develop due to weakening of elastic tissues.
Some subtypes of EDS included within the original disease classification have been redefined and are no longer part of the original nor the revised EDS categorization. For example, what was previously known as EDS type IX has been redefined and is now termed occipital horn syndrome. In addition, EDS type XI is currently known as familial hypermobility syndrome. For more information on these disorders, please see the “Related Disorders” section of this report below.
Most forms of Ehlers-Danlos syndrome (EDS) are transmitted as an autosomal dominant or autosomal recessive trait. Each EDS subtype is a distinct hereditary disorder that may affect individuals within certain families (kindreds). In other words, individuals with one subtype of EDS will not have children with another EDS subtype.
The disease genes that cause some forms of EDS have been mapped to particular chromosomes. Although the specific underlying cause of EDS is not known for all EDS subtypes, the disorder is known to result from various defects of collagen, the major structural protein in the body. Collagen is the tough, fibrous protein that serves to provide elasticity to and strengthen bodily cells and tissues.
EDS classical type is inherited as an autosomal dominant trait. Human traits including the classic genetic diseases, are the product of the interaction of two genes for that condition, one received from the father and one from the mother.
In dominant disorders, a single copy of the disease gene (received from either the mother or father) will be expressed “dominating” the other normal gene and resulting in the appearance of the disease. The risk of transmitting the disorder from affected parent to offspring is 50 percent for each pregnancy regardless of the sex of the resulting child.
According to researchers, in at least some affected individuals, EDS classical type may result from abnormal changes (mutations) in the gene known as collagen type V, alpha-1 (COL5A1), which has been mapped to the long arm (q) of chromosome 9 (9q34.2-q34.3), or the gene collagen type V, alpha-2 (COL5A2), located on the long arm of chromosome 2 (2q31). Chromosomes are found in the nucleus of all body cells. They carry the genetic characteristics of each individual. Pairs of human chromosomes are numbered from 1 through 22, with an unequal 23rd pair of X and Y chromosomes for males and two X chromosomes for females. Each chromosome has a short arm designated as “p” and a long arm identified by the letter “q.” Chromosomes are further subdivided into bands that are numbered.
EDS hypermobility type is transmitted as an autosomal dominant trait. A specific underlying collagen defect responsible for this form of the disorder has not been identified. EDS vascular type is also inherited as an autosomal dominant trait. This subtype is caused by abnormal changes (mutations) of the gene known as collagen type III, alpha-1 (COL3A1), which is located on the long arm of chromosome 2 (2q31).
EDS kyphoscoliosis type is inherited as an autosomal recessive trait. In recessive disorders, the condition does not appear unless a person inherits the same defective gene for the same trait from each parent. If an individual receives one normal gene and one gene for the disease, the person will be a carrier for the disease, but usually will not show symptoms. The risk of transmitting the disease to the children of a couple, both of whom are carriers for a recessive disorder, is 25 percent. Fifty percent of their children risk being carriers of the disease, but generally will not show symptoms of the disorder. Twenty-five percent of their children may receive both normal genes, one from each parent, and will be genetically normal (for that particular trait). The risk is the same for each pregnancy.
In some affected individuals, the kyphoscoliosis subtype is thought to result from mutations of a gene (called “procollagen-lysine, 2-oxoglutarate 5-dioxygenase” [PLOD]) that encodes a collagen-modifying enzyme known as lysyl hydroxylase. Deficiency of this enzyme may result in the symptoms and findings associated with this form of EDS. The PLOD gene has been mapped to the short arm of chromosome 1 (1p36.3-p36.2).
EDS arthrochalasia type is transmitted as an autosomal dominant trait. This subtype may result from mutations of the gene known as collagen type I, alpha-1 (COL1A1), which has been mapped to the long arm of chromosome 17 (17q21.31-q22.05), or the gene called collagen type I, alpha-2 (COL1A2), located on the long arm of chromosome 7 (7q22.1).
EDS dermatosparaxis type has autosomal recessive inheritance. This EDS subtype is thought to be caused by mutations of a gene or genes that encode a collagen-modifying enzyme known as procollagen I N-terminal peptidase.
As discussed above (see “Symptoms”), in addition to the six primary EDS subtypes, there are some other, rare forms of EDS. The rare subtype known as X-linked EDS is, as its name indicates, transmitted as an X-linked trait. X-linked recessive disorders are conditions that are coded on the X chromosome. Females have two X chromosomes, but males have one X chromosome and one Y chromosome. Therefore, in females, disease traits on the X chromosome may be masked by the normal gene on the other X chromosome. Since males only have one X chromosome, if they inherit a gene for a disease present on the X, it will be expressed. Males with X-linked disorders transmit the gene to all their daughters, who are carriers, but never to their sons. Females who are carriers of an X-linked disorder have a 50 percent risk of transmitting the carrier condition to their daughters and a 50 percent risk of transmitting the disease to their sons. In some females who inherit a single copy of a disease gene for an X-linked recessive trait (heterozygotes), disease traits on the X chromosome may not always be masked by the normal gene on the other X chromosome. Therefore, it is possible that some female carriers of the disease gene may exhibit some of the symptoms associated with the disorder; however, according to reports in the medical literature, to date, no female carriers of the disease gene for X-linked EDS have experienced symptoms (asymptomatic carriers).
EDS periodontosis type, another rare subtype, has autosomal dominant inheritance. EDS progeroid form, which is thought to be inherited as an autosomal dominant trait, may be caused by gene mutations that result in deficiency of a particular enzyme (XGPT deficiency). The subtype known as EDS type X (or EDS dysfibronectinemic type), which has been described in several siblings in one affected family (kindred), is thought to have autosomal recessive inheritance.
According to reports in the medical literature, there appear to be additional, rare subtypes of EDS that may have autosomal dominant or autosomal recessive inheritance (e.g., EDS, autosomal dominant, unspecified type; EDS, autosomal recessive, unspecified type).
Males and females are equally affected by autosomal dominant and autosomal recessive forms of Ehlers-Danlos syndrome (EDS). The X-linked subtype of EDS is fully expressed in males only. It is possible that some females who carry a single copy of the disease gene (heterozygotes) for X-linked EDS may develop some symptoms; however, according to the medical literature, reports indicate that no female carriers have developed associated symptoms (asymptomatic).
In many individuals with EDS, associated symptoms and findings may become apparent during childhood. However, depending upon the form of the disorder present, some abnormalities may be apparent at birth. In other cases, such as those with relatively mild disease manifestations, EDS may not be recognized until adulthood.
Reported estimates concerning the disorder’s overall frequency have varied, ranging from one in 5,000 to 10,000 births. However, because those with mild joint and skin manifestations may not seek medical attention or remain undiagnosed, it is difficult to determine the true frequency of EDS in the general population. EDS classical, hypermobility, and vascular types account for most reported cases of the disorder. EDS kyphoscoliosis, arthrochalasia, dermatosparaxis, and other subtypes are considered much less common. For example, some forms of EDS (e.g., EDS type X or EDS dysfibronectinemic type) may have only been reported in individuals within one affected family (kindred).
The first published accounts of Ehlers-Danlos syndrome occurred in 1892. The syndrome was furthered clarified by Ehlers in 1901 and Danlos in 1908.
Some of the symptoms of the following disorders may be similar to those seen in Ehlers-Danlos syndrome (EDS). Comparisons may be useful for a differential diagnosis:
Occipital horn syndrome (OHS), also known as X-linked cutis laxa, is a rare disorder that was formerly classified as a subtype of EDS (EDS type IX). The disorder has been recategorized with other connective tissue diseases that result from defects of copper metabolism. OHS is characterized by abnormally loose skin that tends to hang in folds (cutis laxa); abnormalities of the muscular organ that stores urine (bladder); the formation of “horn-like” bony protuberances on both sides of the back of the skull (occipital horns) and other skeletal abnormalities; excessive extension (hypermobility) of the fingers and toes; and limited extension of the elbows and knees. In some cases, affected individuals may have a prematurely aged facial appearance, a hooked nose, sagging cheeks, downwardly slanting eyelid folds (palpebral fissures), and/or other facial abnormalities. The disorder may also be characterized by mild mental retardation. OHS is transmitted as an X-linked recessive trait and is caused by deficiency of an enzyme (lysyl oxidase deficiency) that results in abnormalities of copper metabolism.
Familial hypermobility syndrome was also formerly categorized as a subtype of EDS (EDS type XI). However, researchers since suggested that the designation of EDS be reserved for the association of joint hypermobility with distinctive skin changes, resulting in the disorder’s separate categorization. Familial hypermobility syndrome is characterized by looseness (laxity) and excessive extension of the joints; recurrent dislocation of certain joints, such as those of the shoulders and knees; and, in some cases, dislocation of the hip joints at birth (congenital). This disorder is transmitted as an autosomal dominant trait.
There are additional disorders that may be characterized by joint hypermobility, skin changes, and/or other abnormalities similar to those associated with EDS, such as other forms of cutis laxa or other related disorders. (For more information on these disorders, please choose “cutis laxa” or other specific disease names as your search term in the Rare Disease Database.)
Ehlers-Danlos syndrome (EDS) is diagnosed based upon a thorough clinical evaluation, characteristic physical findings, a careful patient and family history, and specialized tests.
Specialized diagnostic laboratory tests may be available for certain EDS subtypes in which the specific underlying biochemical defect has been identified and characterized. In addition, in some families (kindreds) affected by a particular EDS subtype who have identified gene mutations, precise genetic testing may be available that enables diagnosis before or after birth (prenatal or postnatal diagnosis). However, it is possible that such testing may only be accessible through research laboratories with a special interest in EDS.
In addition, in some cases, diagnostic testing includes the removal (biopsy) and microscopic examination (e.g., electron microscopy) of small samples of skin tissue. Such examination may reveal characteristic abnormalities in collagen structure seen in certain EDS subtypes.
The clinical evaluation of individuals with suspected or diagnosed EDS typically includes assessments to detect and determine the extent of skin and joint hyperextensibility. For example, physicians may measure skin hyperextensibility by carefully pulling up skin at a neutral site until the point of resistance, and joint hyperextensibility may be evaluated using a clinical rating scale (i.e., Beighton scale). In addition, in some cases, specialized imaging tests, such as computerized tomography (CT) scanning, magnetic resonance imaging (MRI), and echocardiography, are used to detect and characterize mitral valve prolapse and aortic dilatation. During a CT scan, a computer and x-rays create a film showing cross-sectional images of certain bodily structures. MRI uses a magnetic field to create cross-sectional images of particular organs and tissues. During an echocardiogram, sound waves are directed toward the heart, enabling physicians to study cardiac function and motion.
In addition, in some individuals with EDS, specialized x-ray studies may be used to characterize round, movable lumps (calcified spheroids) under the skin; to detect and determine the extent of abnormal spinal curvature (scoliosis and/or kyphosis) and/or reduced bone mass (ostepenia) (e.g., in those with EDS kyphoscoliosis or arthrochalasia types); and/or to confirm and characterize certain other abnormalities.
In some cases, physicians may recommend that individuals with EDS vascular type be monitored with appropriate noninvasive imaging techniques (e.g., CT scanning, MRI, ultrasonography) to ensure early detection of arterial changes (e.g., aneurysms) that may result in spontaneous arterial rupture and potentially life-threatening complications. Angiography, a diagnostic test that is often used to detect aneurysms, must be avoided, since this technique may be hazardous to individuals with EDS, particularly those with EDS vascular type. During angiography, a substance that is impenetrable by x-rays (contrast medium) is injected into an artery via a flexible plastic tube (catheter) and an x-ray series is taken that visualizes blood flow through certain blood vessels.
The treatment of individuals with EDS is directed toward the specific symptoms that are apparent in each individual. Treatment may require the coordinated efforts of a team of specialists who may need to systematically and comprehensively plan an affected individual’s treatment. Such specialists may include pediatricians or internists; specialists who diagnose and treat disorders of the skeleton, joints, muscles, and related tissues (orthopedists); physicians who diagnose and treatment skin disorders (dermatologists); specialists who diagnose and treat connective tissue diseases (rheumatologists); surgeons; physical and occupational therapists; and other health care professionals.
In individuals with EDS, the use of special braces may help to stabilize affected joints. In addition, specialized physical and occupational therapy techniques may help to preserve the joints and strengthen muscles. Parents of young children with the disorder and affected individuals should also take necessary precautions to prevent injuries and trauma, such as may occur during contact sports. Wearing protective clothing and special padding over pressure points (e.g., shins, knees, elbows) may be beneficial.
Females with EDS vascular type should be counseled concerning the increased risk of certain complications during pregnancy and delivery and the need for meticulous obstetric care. In addition, appropriate precautions and careful monitoring are essential before, during, and after dental or surgical procedures. Because fragile tissues and stitched (i.e., sutured) incisions or wounds may easily tear during or after surgery, unnecessary surgical procedures should be avoided. Accordingly, when surgery is necessary in individuals with EDS, specific surgical approaches require careful evaluation.
Genetic counseling will be of benefit for affected individuals and family members. Other treatment for individuals with EDS is symptomatic and supportive.
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This graphic shows clouds of water on Mars.
Click on image for full size
Image from: NASA/JPL
Mars Global Surveyor Measures Water Clouds
You might think that clouds in the sky have to be made of water like those of Earth, but this is not always so. Clouds might be made of carbon dioxide or ammonia. Just take a peek at the planets Jupiter or Venus. This graph, taken by Mars Global Surveyor, shows proof that the clouds of Mars are made of water. The sequence shows the water clouds moving across the face of Mars. The Mariner 9 mission was the first to provide scientists with proof that the clouds of Mars contained water. Mars Pathfinder took images of Martian clouds from the ground level. Clouds seem mostly to be found in the middle of Mars, as the measurement shows. Clouds are also seen at the north pole. This may be because the spring season is approaching, and ice at the north polar is evaporating.
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The Mars Global Surveyor reached Mars in September of 1997. But it didn't make it into its final mapping orbit until February 1999. What took so long? Surveyor needed to reach a near-circular, low-altitude...more
Mars Global Surveyor carries an instrument which measures the heights of things. This instrument is called an altimeter, or "altitude-meter". The graph to the left shows the results returned from Mars...more
Mars Global Surveyor carries an instrument which measures the heights of things. This instrument is called an altimeter, or "altitude-meter". The picture to the left shows Mars Global Surveyor's measurement...more
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A publication of the Archaeological Institute of America
(Click on the map to explore Gran Canaria, or use the links at the bottom of this page.)Gran Canaria, Spain, the most populated of the Canary Islands, is known for its warm climate and golden beaches. Tourists flock to its oceanfront hotels to bask in the sun. But few who visit this island 130 miles from the northwest coast of Africa are aware of its archaeological heritage. Hundreds of caves, occupied from ancient times to the present, hide in its cliffs and mountains. Burial mounds overlook the sea near the remains of stone houses which once sheltered bustling fishing communities. Before tourists came to this island--even before Europeans set foot on its soil--a group of people, most likely from North Africa, made its home there.
The Roman historian Pliny the Elder (A.D. 23-79) called the island Canaria, a reference to the large wild dogs (from the Latin canis, for dog) which he reported living on the island in his 37-volume Natural History:
...[The island is] named Canaria [Gran Canaria], from its multitude of dogs of a huge size [two of these were brought back for Juba*]. [Explorers] said that in this island there are traces of buildings; that while they all have an abundant supply of fruit and of birds of every kind, Canaria also abounds in palm-groves bearing dates and in conifers; that in addition to this there is a large supply of honey, and also papryus grows in the rivers, and sheat-fish; and that these islands are plagued with the rotting carcasses of monstrous creatures that are constantly being cast ashore by the sea.
The native canary, a small brown finch with a poor singing voice, had nothing to do with the naming of the Canary Islands. In fact, the birds received the name from the islands, their native home. The Spaniards caught canaries after the fifteenth-century conquest and brought them to the rest of the world.
In 1402 Jean de Béthencourt, a Norman knight, was sent by Henry III of Castile to take the Canary Islands. He conquered Lanzarote and Fuerteventura, but was defeated by the natives at Gran Canaria. Gran Canaria was, however, the first of the islands to be incorporated under the Spanish crown later in the fifteenth century. The sucessful conquest of the island began in 1478 when General Juan Rejón founded the city of Las Palmas--the first city founded by the Spaniards outside the Spanish mainland--in the island's northeast. The conflict lasted about five years. (See also Bentayga.)
The island became a resting and refueling station for explorers crossing the Atlantic. In 1492, Christopher Columbus stopped on Gran Canaria for repairs before proceeding to the New World. He returned on his second and fourth voyages to resupply his ships.
Amélie A. Walker, online editor and webmaster of ARCHAEOLOGY, thanks the Tourism Office of Gran Canaria for its assistance.Share
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# Brackets & Factors OCR Stage 6. What is 3(2 + 4) ? 3 ‘lots’ of ‘2 + 4’= 3 ‘lots’ of 6= 18 2 4 3 x 6 12 = 18.
## Presentation on theme: "Brackets & Factors OCR Stage 6. What is 3(2 + 4) ? 3 ‘lots’ of ‘2 + 4’= 3 ‘lots’ of 6= 18 2 4 3 x 6 12 = 18."— Presentation transcript:
Brackets & Factors OCR Stage 6
What is 3(2 + 4) ? 3 ‘lots’ of ‘2 + 4’= 3 ‘lots’ of 6= 18 2 4 3 x 6 12 = 18
What is 3(x + 4) ? x 4 3 x 3x12 = 3x + 12 3 ‘lots’ of ‘x + 4’
What is 5(2x + 1) ? 2x 1 5 x 10x5 = 10x + 5 5 ‘lots’ of ‘2x + 1’
What is 4(x - 2) ? x -2 4 x 4x-8 = 4x - 8 4 ‘lots’ of ‘x - 2’
What is x(x + 3) ? x 3 x x x2x2 3x = x 2 + 3x x ‘lots’ of ‘x + 3’ What is x times x?
What is 2x(x - 3) ? x -3 2x x 2x 2 -6x = 2x 2 – 6x 2x ‘lots’ of ‘x - 3’
What is -x(2x + 3) ? 2x 3 -x x -2x 2 -3x = -2x 2 – 3x -x ‘lots’ of ‘2x + 3’ We call this process ‘expanding a bracket’
Factorising an expression This is the reverse of expanding a bracket Eg Factorise 4x + 2 What is a Factor? A FACTOR of a number is a number that divides into the number EXACTLY
Eg Factorise 4x + 2 What number divides into BOTH parts of the expression EXACTLY? 2 4x + 2 2x+1 = 2(2x + 1) How could we check this?
Eg Factorise 6x - 9 What number divides into BOTH parts of the expression EXACTLY? 36x - 9 2x- 3 = 3(2x – 3)
Eg Factorise x 2 + x What number divides into BOTH parts of the expression EXACTLY? xx 2 + x x+ 1 = x(x + 1)
Eg Factorise 8x 2 – 4x What number divides into BOTH parts of the expression EXACTLY? 4x8x 2 – 4x 2x- 1 = 4x(2x - 1)
Eg Factorise 6 - 4x 2 What number divides into BOTH parts of the expression EXACTLY? 26 - 4x 2 3- 2x 2 = 2(3 – 2x 2 )
Download ppt "Brackets & Factors OCR Stage 6. What is 3(2 + 4) ? 3 ‘lots’ of ‘2 + 4’= 3 ‘lots’ of 6= 18 2 4 3 x 6 12 = 18."
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## Question
### Solution
Correct option is
Converting the given in-equations into equations, we obtain the following equations:
3x + 5y = 15, 5x + 2y = 10, x = 0 and y = 0
Region represented by 3x + 5y ≤ 15: The line 3x + 5y = 15 meets the coordinate axes at A(5, 0) and B1 (0, 3) respectively. Join these points to obtain the line 3x + 5y = 15. Clearly, (0, 0) satisfies the in-equation 3x + 5y≤ 15. So, the region containing the origin represents the solution set of the in-equation 3x + 5y ≤ 15.
Region represented by 5x + 2y ≤ 10: The line 5x + 2y = 10 meets the coordinate axes at A2 (2, 0) and B2 (0, 5) respectively. Join these points to obtain the line 5x + 2y = 10. Clearly, (0, 0) satisfies the in-equation 5x + 2y≤ 10. So, the region containing the origin represents the solution set of this in-equation.
Region represented by x ≥ 0 and y ≥ 0: Since every point in the first quadrant satisfies these in-equations. So, the first quadrant is the region represented by the in-equations x ≥ 0 and y ≥ 0.
The shaded region OA2 PB1 in fig. represents the common region of the above in-equations. This region is the feasible region of the given LPP.
The co-ordinates of the vertices (corner-points) of the shaded feasible region are O (0, 0), A2 (2, 0), and B1 (0, 3).
These points have been obtained by solving the equations of the corresponding intersecting lines, simultaneously.
The values of the objective function at these points are given in the following table.
Point (x, y) Value of the objective function Z = 5x + 3y O (0, 0) Z = 5 × 0 + 3 × 0 = 0 A2 (2, 0) Z = 5 × 2 + 3 × 0 = 10 B1 (0, 3) Z = 5 × 0 + 3 × 3 = 9
Clearly, Z is maximum at . Hence, is the optimal solution of the given LPP. The optimal value of Z is
#### SIMILAR QUESTIONS
Q1
A resourceful home decorator manufactures two types of lamps say A andB. Both lamps go through two technicians, first a cutter, second a finisher. Lamp A requires 2 hours of the cutter’s time and 1 hour of the finisher’s time. Lamp B requires 1 hour of cutter’s and 2 hours of finisher’s time. The cutter has 104 hours and finisher has 76 hours of time available each month. Profit on one lamp A is Rs. 6.00 and on one lamp B is Rs 11.00. Assuming that he can sell all that he produces, how many of each type of lamps should he manufacture to obtain the best return.
Q2
A company makes two kinds of leather belts, A and B. Belt A is high quality belt, and B is of lower quality. The respective profits are Rs 4 and Rs 3 per belt. Each belt of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (bothA and B combined). Belt A requires a fancy buckle, and only 400 buckles per day are available. There are only 700 buckles available for belt B. What should be the daily production of each type of belt? Formulate the problem as a LPP.
Q3
A dietician whishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of Vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C while food ‘II’ contains 1 unit per kg of vitamin A and 2 units per kg of vitamin C. It costs Rs 5.00 per kg to purchase food ‘I’ and Rs 7.00 per kg to produce food ‘II’. Formulate the above linear programming problem to minimize the cost of such a mixture.
Q4
A diet is to contain at least 400 units of carbohydrate, 500 units of fat, and 300 units of protein. Two foods are available: F1 which costs Rs 2 per unit, and F2 which costs Rs 4 per unit. A unit of food F1 contains 10 units of carbohydrate, 20 units of fat, and 15 units of protein; a unit of food F2 contains 25 units of carbohydrate, 10 units of fat, and 20 unit of protein. Find the minimum cost for a diet consists of a mixture of these two foods and also meets the minimum nutrition requirements. Formulate the problem as a linear programming problem.
Q5
The objective of a diet problem is to ascertain the quantities of certain foods that should be eaten to meet certain nutritional requirement at minimum cost. The consideration is limited to milk, beaf and eggs, and to vitamins ABC. The number of milligrams of each of these vitamins contained within a unit of each food is given below:
Vitamin Litre of milk Kg of beaf Dozen of eggs Minimum daily requirements A B C 1 100 10 1 10 100 10 10 10 1 mg 50 mg 10 mg Cost Rs 1.00 Rs 1.10 Re 0.50
What is the linear programming formulation for this problem?
Q6
There is a factory located at each of the two places P and Q. From these locations, a certain commodity is delivered to each of the three depots situated at AB and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are 8 and 6 units respectively. The cost of transportation per unit is given below.
To From Cost (in Rs) A B C P Q 16 10 10 12 15 10
How many units should be transported from each factory to each in order that the transportation cost is minimum. Formulate the above as a linear programming problem.
Q7
A brick manufacturer has two depots, A and B, with stocks of 30,000 and 20,000 bricks respectively. He receives orders from three builders PQand R for 15,000, 20,000 and 15,000 bricks respectively. The cost in Rs of transporting 1000 bricks to the builders from the depots are given below:
From To P Q R A B 40 20 20 60 30 40
How should the manufacturer fulfil the orders so as to keep the cost of transportation minimum?
Formulate the above linear programming problem.
Q8
A company is making two products A and B. The cost of producing one unit of products A and B are Rs 60 and Rs 80 respectively. As per the agreement, the company has to supply at least 200 units of product B to its regular customers. One unit product A requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirements. Formulate the problem as a LLP.
Q9
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
Product A Product B Weekly capacity Department 1 3 2 130 Department 2 4 6 260 Selling price per unit Rs 25 Rs 30 Labour cost per unit Rs 16 Rs 20 Raw material cost per unit Rs 4 Rs 4
The problem is to determine the number of units of produce each product so as to maximize total contribution to profit. Formulate this as a LLP.
Q10
Solve the following LPP by graphical method:
Minimize Z = 20x + 10y
Subject to x + 2y ≤ 40
3x + y ≥ 30
4x + 3y ≥ 60
And, xy ≥ 0
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Joint Classification in Human Anatomy
Joints are the connection points between bones. Basically there are movable joints and immovable joints. For training purpose the first category is most interesting. Movable joints are divided into the highly mobile synovial joints, such as knee or shoulder joint, and cartilaginous joints, which allow only little movement, such as joints between sternum and ribs.
Synovial joints have a high range of motion because of a space between the articulating bones, which is filled with synovial fluid. The contact area of each bone is covered with a layer of cartilage, wich is elastically deformable, to protect it from pressure, friction and other forces.
Cartilage belongs to the supportive structures and has no artery. Thatīs why it is recovering or healing very slowly. It can only be nourished and supplied with oxygen by diffusion from surrounding tissues. The premise to that is multilateral movement to have the synoval fluids exchanged, waste products removed and the cartilage filled with nutrients.
Synovial Joint Illustration
Synovial Joint Classification
|Ball-and-Socket-Joint||Distal bone can move around a center in an indefinite number of axes. Main movements are flexion-extension, adduction-abduction, axial rotation and circumduction.||hips, shoulders |
|Ellipsoid Joint||Distal bone has an ovoid articular surface and is received into an elliptical cavity, wich makes it impossible for the bones to do axial rotation. So main movements here are flexion-extension, adduction-abduction and circumduction.||wrist|
|Hinge Joint||Here the distal bone can move only in one plane, flexion and extension (forward and backward). ||knee, elbow, fingers |
|Flat Joint||The main movements are flexion-extension and rotation. ||wrist|
|Saddle Joint||The saddle joint consists of two opposing surfaces that are reciprocally concave-convex, which allows flexion, extension, adduction, abduction, and circumduction, but no axial rotation.||thumb|
|Pivot Joint||A pivot jointīs movement is limited to rotation. ||Atlas and Axis, proximal radioulnar articulation|
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Schooner rigger vessels known for their speed that became famous during the War of 1812 as blockade-runners and privateers and subsequently notorious as slave ships after the slave trade was outlawed by international agreement. Their hulls were long and low with sharp ends and a deep V shape below the waterline. These design features were later incorporated into the true clipper ships that first appeared in the 1840s.
The width of a vessel at its widest point.
Spar that runs along the foot of a fore-and-aft sail.
Front end of a vessel.
Two masted square-rigged vessels.
To turn the vessel so that the bow goes through the eye of the wind.
Also known as tacking as it involves a switch from one tack to the other.
The distance below the water line that a vessel’s keel extends.
A sail that is attached to the mast or stay along its leading edge can carry the wind on either surface depending on the tack and sets parallel to the vessel when in its neutral position. A fore-and-aft rig one that carries that type of sail.
The shorter mast further forward on a schooner rig.
The sail carried on the foremast on a schooner rig.
The sail carried inside and below both the jib topsail and jib and set on a wire cable (stay; see standing rigging) that runs from the bowsprit to the foremast.
The distance between the waterline and the level of the deck on a vessel when it is level.
Spar that runs along the top of a fore-and-aft sail in a traditional schooner rig.
Main body of a vessel.
The sail carried inside and below both the jib topsail and jib and set on a wire cable (stay; see standing rigging) that runs from the jib-boom to the foremast.
The extension of the bowsprit on the bow of a vessel.
The sail carried furthest forward in front of the foremast. It is set on a wire cable (stay; see standing rigging) that runs from the end of the jib-boom to the top of the foremast.
To turn the vessel so that the stern passes through the eye of the wind. The maneuver results in a change of tracks. In a square rigged vessel this maneuver is known as wearing.
Away from the direction that the wind is coming from.
The taller mast that is further back on a schooner rig.
Points of sail
The relative angle of sailing vessel to the wind. See points of sail diagram.
The arrangement of masts and sails on a vessel.
Vertical piece of wood attached to the stern of a vessel, which allows it to be steered.
The lines on a sailing vessel that move either to raise or adjust the sails.
Two masted fore-and-aft rigged vessel generally used in the fisheries and the coastal trade.
Square Fore Topsail
A square sail carried on the foremast below the top gallant on the type of schooner rig common on the Baltimore Clippers.
A sail that is attached to the mast along its center carries the wind on only one surface and sets across the vessel (“square” to the vessel) when in its neutral position. A square rig is one that carries primarily that type of sail (most will carry some fore-and-aft sails as well).
The lines on a sailing vessel (usually wire cable) that hold up the masts. The standing rigging is made up of Stays, which run up to the masts from the forward and back ends of the vessel and Shrouds that run up from the sides of the vessel.
Back of a vessel.
A course defined by which side of the vessel that the wind is on. Stated as either Starboard Tack (wind on the right side of the vessel) or Port Tack (wind on the left side of the vessel). As a verb, to tack means to change course so that the tack changes. (See come about)
The square sail carried at the top of the foremast on the type of schooner rig common on the Baltimore Clippers.
Toward the direction that the wind is coming from.
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Natural variants of crops are generated from wild progenitor plants under both natural and human selection. Diverse crops that are able to adapt to various environmental conditions are valuable resources for crop improvements to meet the food demands of the increasing human population. With the completion of reference genome sequences, the advent of high-throughput sequencing technology now enables rapid and accurate resequencing of a large number of crop genomes to detect the genetic basis of phenotypic variations in crops. Comprehensive maps of genome variations facilitate genome-wide association studies of complex traits and functional investigations of evolutionary changes in crops. These advances will greatly accelerate studies on crop designs via genomics-assisted breeding. Here, we first discuss crop genome studies and describe the development of sequencingbased genotyping and genome-wide association studies in crops. We then review sequencing-based crop domestication studies and offer a perspective on genomics-driven crop designs.
Via Ali Taheri
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Lang: So we are all on to our new reader. After the class discussion on the reader,
plaese complete the following activites in the language diary:
Read the book till page no:15 & prepare a
- List of difficult words and any foreign word that you come across while reading.
- Identify the main character in the book & write a few lines about the same.
- List down the ways in which your life is different from the protagonist or the main character.
Math:Â After the introduction to the coordinates in the class, please look at the following website to clarify any doubts that you may have.Â
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The silica-scaled chrysophytes are unicellular flagellates, photosynthetic or colourless, solitary or colonial, assigned to the class Chrysophyceae. These organisms have species-specific silica structures, which are formed in silica deposition vesicles (SDVs) derived from the Golgi apparatus. The scales do not create a static armour but a dynamic structure that adjusts to the addition of new scales during both cell growth and division. To date, the taxonomy of the silica-scaled chrysophytes is completely based on the morphology of silica scales. The scales vary in size from around 1-10 µm. For accurate identification at the species level, using Electron Microscopy (EM) is essential. Silica-scaled chrysophytes are very special amongst the protists in that they adhere to a species concept, one of morphology, that extends beyond standard protist cell structure, the silica scales giving us an extra morphological criteria on which to base taxonomical efforts.
Members of the silica-scaled chrysophytes are an important part of the freshwater phytoplankton. They could form the majority of phytoplankton in oligotrophic freshwater lakes and ponds. Knowledge of the chrysophyte occurrence and their ecological ranges is useful for monitoring of the status and quality changes of water bodies. Paleolimnological studies have mainly been focused on monitoring eutrophication, acidification and trends in climatic changes. Due to their well-established species identification concept, wide distribution and narrow occurrence spectra, the silica-scaled chrysophytes present themselves as an excellent paleoecological indicator, giving the oportunity to estimate, with more precision, distribution patterns and ecological preferences of particular chrysophyte species. Even so, comprehensive collection of all records about the occurrences of silica-scaled chrysophytes is still needed.
We have established an on-line database where all records dealing with the European silica-scaled chrysophytes can be simply stored and catalogued for reference and use in any subsequent investigations. The database has been developed to store the data about each collection made (including the geographical coordinates, temperature, pH, and conductivity values) and to store a collection of TEM, SEM, or LM microphotographs. For each species listed in the database, the photo gallery, list of all records, and the distribution map are provided, and automatically updated with any new entry. For those species having more than 20 records, the distribution frequencies along the pH, conductivity, and temperature gradients are shown as well.
At the moment, the database holds information about the distribution and ecology of 204 species and infraspecific taxa, based on more than 7500 entries.
We actively invite everybody who is interested in Chrysophyte biogeography, ecology and taxonomy to participate in this database. We have already included published records about the occurrences of silica-scaled chrysophytes in Europe, and will continue to add newly published data to our database. Despite, we will be grateful to everybody who can include its own, even unpublished records or EM pictures.
|new picture||Mallomonas teilingii||Mon, 01/23/2023 - 13:56||admin|
|new record||Mallomonas decora||Tue, 01/10/2023 - 16:41||ynemcova|
|new record||NEMCOVA et al. (2022)||Tue, 01/10/2023 - 16:41||ynemcova|
|new record||Mallomonas decora||Tue, 01/10/2023 - 16:37||ynemcova|
|new record||NEMCOVA et al. (2022)||Tue, 01/10/2023 - 16:37||ynemcova|
|new record||Mallomonas decora||Tue, 01/10/2023 - 16:34||ynemcova|
|new record||Nemcova et al. 2022||Tue, 01/10/2023 - 16:34||ynemcova|
|new record||Mallomonas decora||Tue, 01/10/2023 - 16:31||ynemcova|
|new record||NEMCOVA et al. (2022)||Tue, 01/10/2023 - 16:31||ynemcova|
|new record||Synura papillosa||Tue, 01/10/2023 - 16:17||ynemcova|
|
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Pacing and SBAC
Before Spring Break, we finished Unit 6 and our first look at fractions. With SBAC testing around the corner, we will try to get through as much of Unit 7 (and Unit 5, which we skipped) as we can before we begin testing mid-April. Wish us luck!
Unit 7, Fractions and Decimals
As we are in the middle of an English unit in Literacy, we will continue our work in Math in Spanish for the time being. Unit 7 begins right after Spring Break, April 3rd!
There are two main focuses for this unit, comparing fractions and then our first look at decimals.
1. Comparing Fractions:
A reminder about math homework
As we have more resources for the 2018 curriculum (the English version) than our older Spanish curriculum (it’s being updated next year!), I will not be able to provide different homework for each student while we are in Spanish, although I will differentiate when I can. If a student is struggling with either the difficulty or the amount of math homework, please don’t hesitate to contact me.
I know that the “new math” today can seem confusing and different from what you (and I) learned years ago. To help you navigate this new world with your student, here is a list of the important terms we will be using over the course of Unit 7:
- Common denominator – A common multiple of two or more denominators. (denominador común)
- Decimal number – A representation of a number using the numerals 0 to 9, in which each digit has a value 10 times the digit to its right. A dot or decimal point separates the whole number part of the number on the left from the fractional part on the right. (número decimal)
- Denominator – The number below the bar in a fraction. It shows the total number of equal parts in the fraction. (denominador)
- Equivalent fractions – Two or more fractions that represent the same number. (fracciones equivalentes)
- Fraction – A number that is the sum of unit fractions, each an equal part of a set or part of a whole. (fracción)
- Hundredth – A unit fraction representing one of one hundred parts, written as 0.01 or 1/100. (centésimo)
- Line plot – A diagram that shows the frequency of data on a number line. Also called a dot plot. (diagrama de puntos)
- Mixed number – a number that can be represented by a whole number and a fraction. (número mixto)
- Numerator – The number above the bar in a fraction. It shows the number of equal parts. (numerador)
- Simplify a fraction – To divide the numerator and the denominator of a fraction by the same number to make an equivalent fraction made from fewer but larger unit fractions. (simplificar una fracción)
- Tenth – A unit fraction representing one of ten equal parts of a whole, written as 0.1 or 1/10. (décimo)
- Unit fraction – a fraction whose numerator is 1. It shows one equal part of a whole. (fracción unitaria)
Thank you again for all of your support of student’s math learning at home and at school. We’ve come a long way already this year, and still have so much more to learn!
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# Ch. - 1 Number Systems
## Chapter 1 Ex.1.1 Question 1
Is zero a rational number? Can you write it in the form \begin{align}\frac{p}{q}\end{align} where \begin{align}p \end{align} and \begin{align}q \end{align} are integers and \begin{align}q \ne 0\end{align}?
### Solution
Steps:
Yes, zero is a rational number.
Zero can be written as:
\begin{align}\frac{0}{{{\rm{Any}}\,{\rm{non - zero}}\,{\rm{integer}}}}\end{align}
\begin{align}\text{Example}:\frac{0}{1} = \frac{0}{{ - 2}}\,\,\end{align}
Which is in the form of \begin{align}\frac{p}{q}\end{align}, where \begin{align}p \end{align} and \begin{align}q \end{align} are integers and \begin{align}q \ne 0\end{align}.
## Chapter 1 Ex.1.1 Question 2
Find six rational numbers between $$3$$ and $$4$$.
### Solution
Steps:
We can find any number of rational numbers between two rational numbers. First of all we make the denominators same by multiplying or dividing the given rational numbers by a suitable number. If denominator is already same then depending on number of rational no. we need to find in question, we add one and multiply the result by numerator and denominator.
\begin{align} 3&=\frac{3\times 7}{7}\,\,\,\text{and}\,\,\,\,\,4=\frac{4\times 7}{7} \\ 3&=\frac{21}{7}\,\,\,\,\,\,\,\,\,\text{and }\,\,\,\,4=\frac{28}{7} \\ \end{align}
We can choose $$6$$ rational numbers as:
\begin{align}\frac{22}{7},\frac{23}{7},\frac{24}{7},\frac{25}{7},\frac{26}{7}\,\,\text{and}\,\,\frac{27}{7}\end{align}
## Chapter 1 Ex.1.1 Question 3
Find five rational numbers between \begin{align}\frac{3}{4} \text{ and } \frac{4}{5}\end{align}
### Solution
Steps:
Since we make the denominator same first, then
\begin{align} \frac{3}{4} &=\frac{3 \times 5}{4 \times 5} \\ &=\frac{15}{20} \\ \frac{4}{5} &=\frac{4 \times 4}{5 \times 4} \\ &=\frac{16}{20} \end{align}
Now, we have to find $$5$$ rational numbers.
\begin{align}\therefore\,\frac{15}{20}& =\frac{15\times 6}{20\times 6} \\ &=\frac{90}{120} \\ \frac{16}{20} &=\frac{16\times 6}{20\times 6} \\ & =\frac{96}{120} \\ \end{align}
$$\therefore$$ Five rational numbers between $$\frac{3}{4}$$ and $$\frac{4}{5}$$ are
\begin{align}\frac{91}{120}, \frac{92}{120}, \frac{93}{120}, \frac{94}{120} \text { and } \frac{95}{120} \end{align}
## Chapter 1 Ex.1.1 Question 4
State whether the following statements are true or false. Give reasons for your answers.
### Solution
(i) Every natural number is a whole number.
Steps:
True, because the set of natural numbers is represented as\begin{align}\text{N= {1, 2, 3…….}}\end{align}and the set of whole numbers is \begin{align}\text{W = {0, 1, 2, 3 ………}.}\end{align}We see that every natural number is present in the set of whole numbers. Also, we can see that the as compared to the set of natural numbers, the set of whole numbers contains just one extra number and that number is $$0$$.
(ii) Every integer is a whole number.
Steps:
False. Negative integers are not present in the set of whole numbers.
(iii) Every rational number is a whole number.
Steps:
False. For example \begin{align}\frac{1}{2}\end{align} is a rational number, which is not a whole number.
The chapter 1 starts with an introduction to the number system using some examples (using number lines), followed by exercise problems. Next, the chapter deals with a detailed explanation of irrational numbers followed by coverage of real numbers and their decimal expansion. Later, the chapter explains in detail about the representation of numbers on a number line, followed by operations on real numbers. The solutions to the exercise problems are provided by us and are downloadable in the PDF format.
Number Systems | NCERT Solutions
### math teachers and top
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Related Sections
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# What is a measured number?
## What is a measured number?
Measured Numbers: 1. Numbers obtained by measuring an object with a measuring device such as a ruler, balance, stopwatch, thermometer etc. Significant Figures in Measured Numbers. Significant Figures – digits used to represent a measured number such that only the digit farthest to the right is uncertain.
## What is exact number example?
An exact number is a value that is known with complete certainty. Examples of exact numbers are counted numbers of objects or certain unit conversions. For example, there are exactly 3 feet in 1 yard. There are exactly 12 eggs in a dozen.
Is 1 kg exact or measured?
kilogram (kg), basic unit of mass in the metric system. A kilogram is very nearly equal (it was originally intended to be exactly equal) to the mass of 1,000 cubic cm of water. The pound is defined as equal to 0.45359237 kg, exactly.
### Is 2.0 an exact number?
For example, you report the mass of two moles of carbon atoms. In this case, the “2” is an exact number. It’s written as “2” and not “2.0” or “2.00”. The mass of one mole of carbon atoms is a rounded number (12.01 g/mol) that contains four significant figures.
### Is there an exact measurement?
There is no such thing as a perfect measurement. Even doing something as simple as measuring the length of an object with a ruler is subject to limitations that can affect how close your measurement is to its true value.
What is an exact measurement?
An exact number has absolutely no uncertainty in it. Exact numbers cannot be simplified and have an infinite number of significant figures. Measured numbers have a limited number of significant figures.
## Is 3.14 an exact number?
The symbol is exact; however, the number 3.14 has only three significant figures, while 3.1416 has five….Exact Numbers.
Precision Accuracy
check by repeating measurements check by using a different method
## Is 3lbs an exact number?
As reasoned with the stopwatch, this measure, 3 lbs, is subject to the precision of the scale. It is not an exact number.
Is 1 gallon exact or measured?
the US gallon (US gal) defined as 231 cubic inches (exactly 3.785411784 L), which is used in the US and some Latin American and Caribbean countries; and. the US dry gallon (“usdrygal”), defined as 1⁄8 US bushel (exactly 4.40488377086 L).
### Are exact numbers measured experimentally?
Only experimentally measured numbers or results calculated from them use significant digits. “Find the mass of 0.05 mole of calcium chloride.” The number 0.05 is considered to be an exact number (but the mass is not).
### Is 1000ml 1l exact or measured?
1 milliliter is equivelant to 0.001 liters (one-one thousandth). Therefore, there are 1000 milliliters in a liter: 1000 mL = 1 L. Therefore, anything less than 1000 mL is less than one whole liter, so it must be counted as a decimal.
What is the difference between exact and measured?
Measured vs. Exact Numbers. Exact numbers are numbers that are exact by definition, such as: 1 inch = 2.54 cm or 1 gallon = 231 cubic inches or 1 foot = 12 inches or 1 meter = 100 centimeters. or. numbers that come in integers and are not likely to be available in amounts smaller than integers.
## What kinds of numbers are exact?
Numbers are of two kinds: exact and inexact. An exact number has a value that has no uncertainty associated with it. Exact numbers occur in definitions, in counting, and in simple fractions. An inexact number has a value that has a degree of uncertainty associated with it.
## How many significant figures are there in an exact number?
Certain types of numbers are considered “exact.” For example, there are exactly 16 ounces in one pound. The number 16 would have as many significant figures as needed.
What is an exact number?
Exact Number. Exact number are those that is known with complete certainty and can not be changed. It has an infinite number of significant figures and obtained by counting but they often appear as integers. For example , there are exactly 56 student in computer science department.
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Geometry 5-4 Medians and Altitudes
starstarstarstarstarstarstarstarstarstar
by Matthew Richardson
| 24 Questions
Note from the author:
A complete formative lesson with embedded slideshow, mini lecture screencasts, checks for understanding, practice items, mixed review, and reflection. I create these assignments to supplement each lesson of Pearson's Common Core Edition Algebra 1, Algebra 2, and Geometry courses. See also mathquest.net and twitter.com/mathquestEDU.
The outlined content above was added from outside of Formative.
The outlined content above was added from outside of Formative.
1
2
1
10 pts
Solve It! Draw a large acute scalene △ABC. On each side, mark the point that is ⅕ of the distance from one of the side's endpoints, as shown in the diagram. Connect each fo these points to the opposite vertex. Repeat this process for ¼ and ⅓. What do you think think the results will be for ½? Explain.
2
5 pts
The outlined content above was added from outside of Formative.
3
4
3
10 pts
Problem 1 Got It? In the diagram for Problem 1, ZA = 9. What is the length of segment ZC? Enter only a number. Use decimal form, if necessary.
4
10 pts
Problem 1 Got It? Reasoning: What is the ratio of ZA to AC? Explain.
The outlined content above was added from outside of Formative.
5
5
10 pts
Problem 2 Got It?
A
B
C
D
6
6
10 pts
Problem 2 Got It?
A
B
C
D
7
7
10 pts
Problem 2 Got It?
A
B
C
D
The outlined content above was added from outside of Formative.
8
8
10 pts
Problem 3 Got It?
A
B
C
D
9
9
10 pts
A
B
10
10
10 pts
A
B
C
D
11
11
10 pts
A
B
C
D
12
12
10 pts
A
B
C
D
13
14
10 pts
Reasoning: Does it matter which two altitudes you use to locate the orthocenter of a triangle? Explain.
15
10 pts
Reasoning: The orthocenter of △ABC lies at vertex A. What can you conclude about segment BA and segment AC? Explain.
16
16
10 pts
Review Lesson 5-3: Is segment XY a perpendicular bisector, an angle bisector, or neither?
perpendicular bisector
angle bisector
neither
17
17
10 pts
Review Lesson 5-3: Is segment XY a perpendicular bisector, an angle bisector, or neither?
perpendicular bisector
angle bisector
neither
18
10 pts
Review Lesson 2-2: Match the statemens on the left with their negations on the right. Each statement should only be used once.
• Two angles are congruent.
• You are not 16 years old.
• m∠A < 90
• m∠A > 90
• Two angles are not congruent.
• You are 16 years old.
• m∠A ≥ 90
• You are 17 years old.
19
10 pts
Vocabulary Review: Use Yes and No to respond to each question.
• Yes
• No
• Are three diameters of a circle concurrent?
• Are two diagonals of a rectangle concurrent?
• Is the intersection of three streets a point of concurrency?
20
10 pts
Use Your Vocabulary: Label each statement as True or False.
• True
• False
• The median of a triangle is a segment that connects the midpoint of one side to the midpoint of an adjacent side.
• The point of concurrency of the medians of a triangle is where they intersect.
• A triangle has only one median.
21
22
23
21
5 pts
GE
HC
EF
DJ
22
5 pts
Your response should be in the form of two adjacent letters and not contain spaces.
23
5 pts
⅔(CH)
⅔(FH)
CH
⅔(EG)
24
10 pts
Reflection: Math Success
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## Calculus: Early Transcendentals (2nd Edition)
Published by Pearson
# Chapter 1 - Functions - 1.4 Trigonometric Functions and Their Inverse - 1.4 Exercises - Page 48: 34
#### Answer
$= - \sec x$
#### Work Step by Step
$\begin{gathered} Using\,\,the\,trigonometric\,identity\,for\,the\,cosine\,of\,a\,sum \hfill \\ \cos \,\,\left( {a + b} \right) = \cos a\cos b - \sin a\sin b \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( {x + \pi } \right)}} \hfill \\ \hfill \\ {\text{Simplify}} \hfill \\ \hfill \\ \sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( x \right)\cos \,\left( \pi \right) - \sin \,\left( x \right)\sin \,\left( \pi \right)}} \hfill \\ \hfill \\ \sec \,\left( {x + \pi } \right) = \frac{1}{{\cos \,\left( x \right) \cdot \,\left( { - 1} \right) - \sin \,\left( x \right) \cdot 0}} = \frac{1}{{ - \cos \,\left( x \right)}} = - \sec x \hfill \\ \hfill \\ which\,was\,needed\,to\,be\,shown. \hfill \\ \end{gathered}$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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NASA's Goddard Institute for Space Studies in New York released the new analysis based on satellite observations, meteorological stations and Antarctic research station observations. The record "is one of several global temperature analyses," the NASA release says, "along with those produced by the Met Office Hadley Centre in the United Kingdom and the National Oceanic and Atmospheric Administration's National Climatic Data Center in Asheville, N.C. These three primary records use slightly different methods, but overall, their trends show close agreement."
The new analysis gives a nod to global warming skeptics -- saying that weather patterns always cause temperature variations from year to year -- but says that if the current emission of greenhouse gases remains the same, each decade in the future will be warmer than the one before it. Greenhouse gases, including carbon dioxide generated by burning fossil fuels, allow sunlight to penetrate the atmosphere but trap its heat and do not allow it to escape back into space. NASA says the carbon dioxide level in the atmosphere was 280 parts per million in 1880 and exceeds 390 ppm now. The average global temperature since 1880 has risen 1.4 degrees Fahrenheit, the analysis says.
"One more year of numbers isn't in itself significant," Goddard climatologist Gavin Schmidt said in a NASA press release. "What matters is this decade is warmer than the last decade, and that decade was warmer than the decade before. The planet is warming. The reason it's warming is because we are pumping increasing amounts of carbon dioxide into the atmosphere."
The National Oceanic and Atmospheric Administration has already reported that 2012 was the warmest summer ever for the continental United States. Goddard Director Dr. James Hansen agreed it is part of a trend. "The climate dice are now loaded," Hansen said in a statement. "Some seasons still will be cooler than the long-term average, but the perceptive person should notice that the frequency of unusually warm extremes is increasing. It is the extremes that have the most impact on people and other life on the planet."(Follow me on Twitter @leeroop)
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GMAT数学练习题及答案【9】!在GMAT数学备考中考生一定要结合练习掌握考试知识点，为了方便考生做练习，智课网为大家整理了GMAT数学练习题及答案，更多GMAT数学题库资料尽在智课网。
If x and kare integers and (12^x)[4^(2x+1)] = (2^k)(3^2),what is the value of k?
A. 5
B. 7
C. 10
D. 12
E. 14
参考答案：E
题目翻译：有等式(12^x)[4^(2x+1)] = (2^k)(3^2)，其中k和x都是整数，求k。
题目全解：等式右边=[4^(k/2)]*(3^2)=[4^(k/2-2)](12^2);
因此，x=2，2x+1=k/2-2;
k=14.
If(y+3)(y-1) – (y-2)(y-1) = r(y-1), what is the value of y?
(1) r^2 = 25
(2) r = 5
A.Statement (1) ALONE is sufficient, but statement (2) alone is notsufficient.
B.Statement (2) ALONE is sufficient, but statement (1) alone is notsufficient.
C. BOTH statementsTOGETHER are sufficient, but NEITHER statement ALONE issufficient.
D. EACHstatement ALONE is sufficient.
E.Statements (1) and (2) TOGETHER are NOT sufficient.
参考答案：E
题目翻译：有等式 (y+3)(y-1) – (y-2)(y-1) = r(y-1), 求y值。
题目全解：左边等于y^2+2y-3-y^2+3y-2=5y-5=5(y-1)
如果5(y-1)=r(y-1),即(5-r)(y-1)=0;
如果r=5,y值不可求;r!=5,y值为1。所以选E。注意，条件(1)，r等5或者-5，y值不确定。
What isthe sum of the integers from -190 to 195, inclusive?
A. 0
B. 5
C. 375
D. 875
E. 965
参考答案：E
题目翻译：负190到正195，整数的和是多少呀?
题目全解：-190到190都抵消了，答案是191+192+193+194+195=965.
GMAT数学相关练习：
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Images Spanning 20 Years Reveal Disappearing Nebula
Great things take time. This is true when it comes to many processes in the universe. For example, it takes millions of years for stars—the building blocks of the universe—to form. Then, many stars last for billions of years before they die and begin to eject shells of gas that glow against the vastness of space—what we call nebulas. It can be exceedingly rare to capture some of these processes in real time.
Lucky for us, it seems as if the Stingray nebula, Hen 3-1357, was destined to stand out from the crowd since its beginnings. It was dubbed the youngest known planetary nebula in 1998 after Hubble caught a rare peek at the central star’s final stages of life. Now, twenty years after its first snapshot, the Stingray nebula is capturing the attention of astronomers again for a very different reason.
Images from 2016 show a nebula that has drastically faded over the last two decades. Additionally, shells of gas that surrounded the central star have changed, no longer as crisp as they once were. Changes like this have never been captured at this clarity before.
Astronomers have caught a rare look at a rapidly fading shroud of gas around an aging star. Archival data from NASA’s Hubble Space Telescope reveal that the nebula Hen 3-1357, nicknamed the Stingray nebula, has faded precipitously over just the past two decades. Witnessing such a swift rate of change in a planetary nebula is exceeding rare, say researchers.
Images captured by Hubble in 2016, when compared to Hubble images taken in 1996, show a nebula that has drastically dimmed in brightness and changed shape. Bright blue fluorescent tendrils and filaments of gas toward the center of the nebula have all but disappeared, and the wavy edges that earned this nebula its aquatic-themed name are virtually gone. The young nebula no longer pops against the black velvet background of the vast universe.
“This is very, very dramatic, and very weird,” said team member Martín A. Guerrero of the Instituto de Astrofísica de Andalucía in Granada, Spain. “What we’re witnessing is a nebula’s evolution in real-time. In a span of years, we see variations in the nebula. We have not seen that before with the clarity we get with this view.”
Researchers discovered unprecedented changes in the light emitted by glowing nitrogen, hydrogen and oxygen being blasted off by the dying star at the center of the nebula. The oxygen emission, in particular, dropped in brightness by a factor of nearly 1,000 between 1996 and 2016.
“Changes in nebulae have been seen before, but what we have here are changes in the fundamental structure of the nebula,” said Bruce Balick of the University of Washington Seattle, leader of the new research. “In most studies, the nebula usually gets bigger. Here, it’s fundamentally changing its shape and getting fainter, and doing so on an unprecedented time scale. Moreover, to our surprise, it’s not growing any larger. Indeed, the once-bright inner elliptical ring seems to be shrinking as it fades.”
Ground-based observations of other planetary nebulae have shown hints of changes in brightness over time, but those speculations haven’t been confirmed until now. Only Hubble can resolve the changes in structure in this tiny nebula. The new paper examines every image of the Stingray nebula from Hubble’s archives.
“Because of Hubble’s optical stability, we are very, very confident that this nebula is changing in brightness with time,” added Guerrero. “This is something that can only be confirmed with Hubble’s visual acuity.”
The researchers note the nebula’s rapid changes are a response to its central star, SAO 244567, expanding due to a temperature drop, and in turn emitting less ionizing radiation.
A 2016 study by Nicole Reindl now of the University of Potsdam, Germany, and a team of international researchers, also using Hubble data, noted the star at the center of the Stingray nebula, SAO 244567, is special in its own right.
Observations from 1971 to 2002 showed the temperature of the star skyrocketing from less than 40,000 to 108,000 degrees Fahrenheit, more than ten times hotter than the surface of our Sun. Now, Reindl and her research team has shown that SAO 245567 is cooling. Reindl speculates the temperature jump was caused by a brief flash of helium fusion that occurred in a shell around the core of the central star. Recently, the star appears to be backstepping into its early stage of stellar evolution.
“We’re very lucky to observe it just in that moment,” said Reindl. “During such a helium shell flash, it evolves very quickly and that implies short evolutionary timescales so we can’t usually see how these stars evolve. We just happened to be there at the right time to have caught that.”
The team studying the rapid fading of the Stingray nebula can only speculate at this time what’s in store for the future of this young nebula. At its present rates of fading, it’s estimated the nebula will barely be detectable in 20 or 30 years.
The Hubble Space Telescope is a project of international cooperation between NASA and ESA (European Space Agency). NASA’s Goddard Space Flight Center in Greenbelt, Maryland, manages the telescope. The Space Telescope Science Institute (STScI) in Baltimore, Maryland, conducts Hubble science operations. STScI is operated for NASA by the Association of Universities for Research in Astronomy in Washington, D.C.
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Work%2C+Energy%2C+and+Power
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Work, Energy Concept:
Work, Energy Concept Made by :Abhaygoyal
James Joule:
James Joule British physicist James Joule is best known for his work in electricity and thermodynamics Together with the physicist William Thomson (later Baron Kelvin), Joule found that the temperature of a gas falls when it expands without doing any work. This principle, which became known as the Joule-Thomson effect, underlies the operation of common refrigeration and air conditioning systems. The metric system unit of energy is the joule (J), after James Joule.
Mechanical:
Mechanical Mechanical energy is the energy which is possessed by an object due to its motion or its stored energy of position Kinetic energy : is the energy of motion Potential Energy : an object can store energy as the result of its position or elastic source
Work Concept:
Work Concept Work is defined as a force acting upon an object to cause a displacement Mathematically, work can be expressed by the following equation. W= F x d cos q ( cos 0 0 = 1) where F = force, d = displacement, and the angle (theta) is defined as the angle between the force and the displacement vector
Work Calculations:
Work Calculations W=F x d W=F x d cos 30 0 W= F x d =100N X 5m = 100N X 5m X .87 =15Kg(10m/s 2) X 5m =500 N m = 413 N m = 750 N m
Gravitational Potential Energy:
Gravitational Potential Energy After an object has been lifted to a height, work is done. PE = W= F x d= mah Potential Energy is maximum at the maximum HEIGHT
Potential Energy Calculation:
Potential Energy Calculation How much potential energy is lost by a 5Kg object to kinetic energy due a decrease in height of 4.5 m PE = mah PE = (5Kg)(10 m/s 2 )(4.5 m) PE = 225 Kg m 2 /s 2 PE = 225 J
Kinetic Energy Calculation:
Kinetic Energy Calculation The energy of motion D KE = W= F x d= mah=1/2 mv 2 Find the kinetic energy of an 4 Kg object moving at 5m/s. KE = 1/2 mv 2 KE = ½ (4Kg)(5m/s) 2 KE = 50 Kg m 2 /s 2 KE = 50 J
Spring constant Calculation:
Spring constant Calculation A tired squirrel (mass of 1 kg) does push-ups by applying a force to elevate its center-of-mass by 5 cm. (A) Determine the number of push-ups which a tired squirrel must do in order to do a mere 5.0 Joules of work. (B) Determine the squirrel’s spring constant.
Spring Constant Calculation:
Spring Constant Calculation W = F x d = 10 N*(.05m)=.5 N m W = .5 J (each push up) 10 pushups = 5 J PE = ½ k x 2 .5 J = ½ k (.05m) 2 .5 J = ½ k (.003m 2 ) .5 J = .0015 m 2 333.3 J/m 2 = k
Power!:
Power! Power is the rate that we use energy. Power = Work or Energy / Time P = W/t = F x d/t = F v The units for power : J/s Kg m 2 / s 2 /s N m / s
Power Calculation:
Power Calculation A 5 Kg Cart is pushed by a 30 N force against friction for a distance of 10m in 5 seconds. Determine the Power needed to move the cart. P = F x d / t P = 30 N (10 m) / 5 s P = 60 N m /s P = 60 watts
Summary:
Summary Energy is the ability to move Potential is stored energy (Statics) Dependant on height Kinetic is moving energy (Dynamics) Dependant on velocity Springs store energy dependant on distance and constant
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Gulf oil spill's environmental impact: How long to recover?
What scientists know about how oil spills affect the environment is drawn from a range of past events, no two of which have been alike. Because the blowout occurred 5,000 feet below below the water surface, the Gulf oil spill is unchartered territory.
Grand Isle, La.; and Boston
Occasionally, he wades into the water to get a closer look at seabirds bobbing and drifting on the sea surface. He returns to the sand, shells cracking under his boots, and says that these patrols are “not something we do on a normal basis.”
But these have not been normal times for Mr. Fischer, director of the state’s new marine biology lab on Grand Isle.
The April 20 undersea oil blowout that destroyed the Deepwater Horizon oil rig and killed 11 oil workers some 40 miles offshore has spewed more than 3.5 million gallons of oil into the Gulf so far. And efforts to slow or halt the 200,000 gallon-a-day flow have failed to this point.
On Saturday, tar balls as big as golf balls began washing ashore on Alabama's Dauphin Island, a barrier island that helps protect the entrance to Mobile Bay and some 16 miles of coastline to the west.
The blowout has been “a new challenge for everyone, for all academia, the science community, the universities,” Fischer says.
What scientists know about how oil spills can affect the environment – and for how long – is drawn from a range of past events, no two of which have been alike. It means that “the leading scientists can build a model for what they think is going to happen, but we may wake up the next morning and not know exactly what to expect,” says Fischer.
Comparisons with the Exxon Valdez spill, for example, can be misleading because of significant differences in the type of oil, the ecosystems affected, and the way natural processes break down oil.
Moreover, the uncertainty is greater in the current spill. For the most part, researchers have studied the aftermath of surface spills. The Deepwater Horizon blowout occurred at 5,000 feet, dispensing crude oil from seafloor to surface.
Many of the long-term effects may remain hidden as natural processes and chemical dispersants break up the oil into small globules dense enough to sink to the bottom. There, it has the potential to affect bottom dwellers for decades.
Dr. McDowell was a coauthor of a 2003 National Academy of Sciences report that remains a seminal work for understanding the behavior of petroleum and petroleum products spilled in marine and coastal environments, many marine scientists say.
For Louisiana in particular, a key area of concern is coastal marshes. They are the breeding ground as well as home base for a wide range of marine life vital to the region’s fishing industries. Moreover, the wetlands provide a first barrier against storm surges from hurricanes.
But southern Louisiana’s wetlands already are stressed – vanishing as the Mississippi Delta sinks beneath the ocean at a rate that, by some estimates, averages 50 acres a day. In addition, the fisheries off the coast are exposed to an annual “dead zone” each spring as nutrient-rich water from the continental heartland moves down the Mississippi and into the Gulf, triggering algae blooms. When the algae die and decompose, the process uses up much of the dissolved oxygen in the water. Fish flee, but bottom dwellers – crabs and other shellfish – generally can’t move fast enough to do so.
If the blowout “turns into something that takes months to shut off ... that is our biggest concern,” says James Cowan Jr., a fisheries ecologist at Louisiana State University in Baton Rouge. With the ecosystem already distressed, “We are concerned it may be at a tipping point.”
In trying to assess the potential effect of oil on the Gulf Coast wetlands, a 1969 spill in Massachusetts’ Buzzards Bay might offer close – if still imperfect – parallels, say Dr. McDowell and Woods Hole colleague Christopher Reddy.
It opens a window on the biological processes that over time help ease the effects of the spill. But it also highlights the long-term effects that can remain after much of the surface evidence has vanished.
The spill involved the barge Florida, which ran aground, dumping 175,000 gallons of diesel fuel. The first organisms to recolonize the area were “opportunistic species” such as carbon-loving worms and microbes, McDowell says. As they ate up their carbon-rich food source – in effect cleansing the harbor of much of the hydrocarbons – they died off, making way for species that normally inhabited the harbor and its marshland to return.
Still, she says, it took about a decade before researchers began to see the kinds of organisms in the harbor one would have seen prior to the spill, such as fiddler crabs. Dr. Reddy has continued to take samples from the marsh, and some 40 years later, oil remains trapped in the sediment three to eight inches below the surface. And it has changed little chemically since its arrival.
“If you stick a shovel into the ground and lift it, you will smell diesel fuel,” he says. “And when you analyze it, it doesn’t look like it’s been significantly changed chemically. And the fiddler crabs, mussels, and marsh grasses are not as healthy” as they are at pristine sites.
Yet above ground, he adds, the marsh looks to have recovered to the point where it could grace a tourist’s postcard.
Another nearby site affected by a spill in 1974 and still under study has not fared as well. In many places, the marsh grass still hasn't returned. And when a research team looked at aerial photos of the site taken before the spill, the researchers found evidence of post-spill erosion in areas that received the most oil. The oil worked its way into the soil, killing off marsh grasses that would have stemmed the erosion.
Reddy cautions that the impact of oil spills depends a great deal on location and dose. Differences in air and ocean temperatures can play a significant role in the pace at which biological processes can begin to blunt the effects of oil.
Research efforts such as this represent a cautionary tale of another sort, according to Joanna Burger, a Rutgers University ecologist. Beware of the small numbers, as in: Only 20 percent of the wetlands have been affected.
“The problem is it’s always the 20 percent of the marsh that’s on the edge, in the intertidal zone,” she says. “That’s the most productive zone in terms of invertebrates and small fishes. And it’s where the herons and egrets feed. You might have destroyed only 20 percent of the marsh, but you might have destroyed 90 percent of the animal production.”
The point is not lost on Louisiana’s Fischer: “As we lose the coastal edge, we are losing the productivity of the area. Throughout history we have had various types of small tragedies such as freezes and pollutants, but we’ve never experienced a large case of oil intrusion into the estuarine areas. What would happen? I don’t have the answer.”
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Food deserts have been a big topic in the United States lately, and for good reason: 23.5 million Americans, about 2% of all households, live in food deserts. These food deserts are defined as areas where it is impossible to find adequate fresh, whole, and healthy foods, particularly fruits and vegetables.
Most food desert areas are impoverished, and many residents of these areas do not have a car to travel to stores that carry healthy options. Not even large cities are exempt from food deserts—according to a 2009 survey, 750,000 New York residents lived in areas without adequate access to healthy food, and in 2006, 500,000 Chicago residents reported the same.
Obviously, having poor access to healthy food has health ramifications. Many people with dietary restrictions like gluten allergies or lactose intolerance have an even harder time getting proper nutrition. It is estimated that healthy eating could save $71 billion in chronic healthcare costs by lowering rates of certain diseases. In neighborhoods with access to healthy food, a 45% decrease in diabetes cases over the course of five years was noted. Obesity was also lower in these areas.
Health costs aside, there are other negative ramifications of living in a food desert. Healthy food is more expensive at smaller stores, and many people don’t have time to venture out to larger stores, even if they do have transportation. Since many people who live in these areas live in poverty, they often have no choice but to eat fast food.
The good news is that there are many different initiatives working toward fewer food deserts in the United States, including school lunch programs and bringing grocery stores to impoverished areas. Learn more from this Tulane University infographic from their School of Social Work.
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# Multiplying and Dividing Fractions
Introduction
Lower Elementary Montessori students learn to multiply and divide fractions by whole numbers. These operations are similar to multiplying and dividing whole numbers.
Multiplying Fractions
To multiply a fraction by a whole number, the numerator is simply multiplied by the whole number. For example, to multiply 2/9 by 4, the numerator 2 is multiplied by the whole number 4, for a product of 8/9.
Montessori students first learn this concept by manipulating Fraction Circles. It is best to start with prepared equations with products less than or equal to one. This way, the products fit in a single Fraction Circle and are easier to visualize.
Dividing Fractions
To divide a fraction by a whole number, the numerator is simply divided by the whole number. For example, to divide 8/9 by 4, the numerator 8 is divided by the whole number 4, for a quotient of 2/9. Montessori students first learn this concept by manipulating Fraction Circles. For example, to divide 8/9 by 4, the student separates the eights fraction pieces into four equal groups. The student then sees that each of the four groups contains two pieces.
The amount to be divided is called the dividend, and the result is called the quotient. Then introducing the concept of dividing fractions by whole numbers, it is easiest to use prepared equations with dividends and quotients less than or equal to one. This way, each equation can be completed using a single Fraction Circle.
The number by which the dividend is being divided is called the divisor. For this introduction, it is necessary to choose equations in which the divisor divides evenly into the numerator.
Not all equations are as simple as those in the above examples. It is sometimes necessary to express a fraction in different terms so the divisor will divide evenly into the numerator. For example, the equation 3/5 ÷ 2 = can be expressed as 6/10 ÷ 2 = . Two divides evenly into 6, giving a quotient of 3/10 for the equation.
Montessori students learn this concept by working with Fraction Circles and experimenting until they find an equivalent fraction that can be divided evenly. To do this, the student needs prepared equations that can easily be worked out using Fraction Circles.
Multiplying Fractions by Whole Numbers
Purpose:
To learn how to multiply fractions by whole numbers.
Material:
Fraction Circles
Prepared Equation Slips
Math journals and pencils
Presentation:
Most Montessori teachers present this concept in Year 3.
Setting Up the Equation
– Invite a student to learn to multiply fractions by whole numbers at a mat where the material is already laid out.
– Encourage the student to choose an equation slip and read it, for example (3/10 x 3 = .
– Tell the student that the equation means taking three tenths three times.
– Ask the student to write the equation in his/her journal.
– Invite the student to take the tenths frame and place it on the mat.
Solving the Equation
– Encourage the student to take three tenths and place them on the mat to the right of the frame.
– Invite the student to repeat this twice more, leaving a space between each group of three. She/he will end up with three groups of three tenths.
– Invite the student to count the fraction pieces on the mat and say how many there are. (Nine.)
– Say to the student that three tenths taken three times equals nine tenths. In other words, three tenths times three equals nine tenths.
– Ask the student to replace the tenths in their frame on the board.
– Invite the student to continue in the same manner with anther question.
– Encourage the student to do several equations until she/he is proficient at multiplying fractions by whole numbers.
Extension:
Invite a student to copy a prepared equation into his/her journal and find the answer without using the Fraction Circles. Repeat for other prepared equations.
Dividing Fractions by Whole Numbers
Purpose:
To learn how to divide fractions by whole numbers.
Material:
Fraction Circles
Green Skittles
Prepared Equation Slips
Math journals and pencils
Presentation
Most Montessori teachers present this concept in Year 3.
Setting Up the Equation
– Invite a student to learn to divide fractions by whole numbers at a mat where the material is already laid out.
– Invite the student to choose an equation slip and read it, for example 6/8 ÷ 2 = .
– Ask the student to copy the equation in his/her journal.
– Review the terms dividend and divisor with the student.
– Encourage the student to make the dividend 6/8 at the center of the work area by placing six eighths pieces together on the mat.
– Ask the student to select enough skittles to make the divisor and place them to the right of the dividend on the mat. (Two skittles for the example of 6/8 ÷ 2 = .)
Solving the Equation
– Remind the student that division means sharing equally.
– Encourage the student to begin sharing equally, placing one fraction piece under each skittle until they are all shared.
– Remind the student that the amount one skittle receives is the answer (quotient) to the division question.
– Ask the student what the quotient is for 6/8 ÷ 2 = . (It is 3/8.)
– Encourage the student to place the eighths pieces back in their frame on the board
– Encourage the student to continue in the same manner with another equation.
Extension:
Invite a student to select a prepared equation slip and find the answer without using Fraction Circles. Repeat for some other prepared equations.
Dividing Fractions by Whole Numbers When Equivalent Fractions Must Be Made First
Purpose:
To learn how to divide fractions by whole numbers when making equivalences is necessary prior to solving equations.
Material:
Fraction Circles
Green Skittles
Prepared Equation Slips
Math journals and pencils
Presentation:
Most Montessori teachers present this concept in Year 3.
Setting Up The Equation
– Invite the student to choose an equation slip and read it, for example, 1/2 ÷ 3 = .
– Ask the student to copy the equation in her/his journal.
– Invite the student to make the dividend 1/2 in the center of the work area by placing a halves piece on the mat.
– Invite the student to select the right number of Green Skittles to make the divisor and place them to the right of the dividend on the mat. (Three skittles.)
– Ask the student what he/she notices about the dividend (One half can not be equally shared among three skittles.)
Finding the Equivalent Fraction and Solving the Equation
– Encourage the student to find a fraction that is equivalent to 1/2 that can be shared equally among three skittles. Coach the student as necessary until she/he finds an equivalence.
– Once the student has found the right equivalence (3/6), ask him/her to record the new equation in the math journal as follows: 3/6 ÷ 3 = . Encourage the student to return the halves inset to the board.
– Encourage the student to begin sharing fraction pieces equally among the three skittles, and stop when the fraction pieces have been used up.
– Remind the student that the amount one skittle receives is the answer to the equation.
– Ask the student what the quotient is. (One sixth.)
– Ask the student to complete the equation in his/her math journal as follows: 3/6 ÷ 3 = 1/6.
– Encourage the student to return the sixth pieces to their frame on the board.
– Invite the student to continue in the same manner with more equations until he/she is proficient at dividing fractions that require equivalences.
Extensions:
Invite a student to imagine he/she has one-third of a chocolate bar to share with one friend. How much chocolate will each person get? Invite the student to find the answer using Fraction Circles and Skittles.
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Researchers are already aware of the potential benefits of electronic devices that send and receive digital pulses at frequencies in the terahertz region of the electromagnetic spectrum. Devices for airport security, medical imaging, drug and food inspection, and high-speed communication, will be much more sensitive than today’s versions—that is, if researchers can develop better sources and detectors for that type of radiation. Now a team of scientists at the University of Maryland reports that it has used graphene to build a terahertz device that is at least as sensitive, and many times as fast, as existing detectors.
Graphene, a sheet of carbon atoms only one atomic layer thick, works well as a terahertz detector because of its ability to absorb radiation, from the ultraviolet to the terahertz regions, equally well. Meanwhile, terahertz radiation, also known as T-rays, can penetrate a wide variety of materials without the ionizing effects of x-rays, and can spectrographically identify materials, making it ideal for applications such as identifying drugs or explosives without harming people.
“[The graphene-based device] as good as any room-temperature detector in this spectral range, and potentially much better,” says Dennis Drew, a research scientist at the University of Maryland’s Center for Nanophysics and Advanced Materials. Drew and his colleagues presented their findings in the latest issue of Nature Nanotechnology.
The detector relies on the photothermoelectric effect. Photons striking the graphene cause electrons in the material to jump to a higher energy level. The affected graphene molecules want to dissipate the resulting thermal energy, but because the electrons lose the heat to the surrounding molecules rather slowly, placing metal contacts on the graphene allows the material to shed excess energy by pushing electrons to the metal. If the contacts are made of two different metals with different conductivity—in this case, gold and chromium—the result is a current. Measuring the current reveals how much terahertz power is being absorbed by the graphene.
Drew says the new detector is as sensitive as the Golay cell, another device used to detect terahertz rays. But while the Golay cell has a response time on the order of a second, the graphene detector makes the measurement in 0.1 nanosecond. Another alternative, a pyroelectric detector, has response times measured in milliseconds, and tends to be somewhat less sensitive.
The graphene detector’s ability to pick up terahertz rays might be further improved by various means, Drew says. Using multiple layers of the material may allow it to capture more radiation. Adding voltage gates to create P-N junctions could also raise such a detector’s performance. Contacts made from metals other than the ones used in the experiments detailed in the paper—aluminum, for example—might also increase the efficiency, though it’s harder to get aluminum to adhere to graphene. Drew says optimizing the performance is a relatively easy engineering challenge.
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The liver is the largest internal organ in the body and weighs about 1.5 kg.
It’s surrounded by a capsule of fibrous connective tissue called Glisson’s capsule.
If we look at the liver from an inferior view, which is a view from the bottom of the liver, we can see that the liver is divided into a large left lobe and right lobe, as well as two smaller lobes, called the quadrate and caudate lobes.
The liver parenchyma or functional tissue of the liver is organized into thousands of hepatic lobules, which have a dual blood supply that comes from terminal branches of the hepatic portal vein and hepatic artery.
The blood then flows through sinusoids surrounded by hepatocytes before draining into the lobule’s central vein.
Hepatocytes are the main functional cells of the liver that perform a large variety of functions, including the production of bile, a number of plasma proteins, and non-essential amino acids; the metabolism of fat, carbohydrate, and protein; the storage of glucose, vitamins, and iron; and the breakdown or detoxification of metabolic waste products, drugs, and toxins.
After identifying the lobule, it can be easier to locate portal triads in an image since they’re typically located at the corners of the lobules.
If we take a closer look at just one portal triad, we can more easily identify the portal venule by its large diameter and thin walls compared to the arteriole, which has a much smaller diameter and thicker walls.
Similar to this image, the portal tract can sometimes have more than one bile duct.
The bile ducts can be identified by their prominent simple cuboidal epithelium.
Let’s now take a closer look at the hepatocytes, which are large polygonal epithelial cells that form branching plates that are only one-cell thick, separated by sinusoids, and radiate outward from the central vein.
The sinusoids carry blood from the hepatic arteriole and portal venule to the central vein, while the bile canaliculi or capillaries carry the bile produced by hepatocytes in the opposite direction in order to drain into the bile ductules.
The hepatocyte’s cytoplasm is very eosinophilic, or pink because they contain a lot of mitochondria.
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# How do you factor 2d^3-d^2-3+6d?
Apr 29, 2015
You may take a close look at the co-efficients.
Some rearranging:
$= \left(2 {d}^{3} - {d}^{2}\right) + \left(6 d - 3\right)$
Then we take common things out of the brackets:
$= {d}^{2} \left(2 d - 1\right) + 3 \left(2 d - 1\right)$
Since inside the brackets is now the same, we may go:
$= \left({d}^{2} + 3\right) \left(2 d - 1\right)$
${d}^{2} + 3$ cannot be factored further.
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# Bresenham’s Algorithm
Written by
Jack Bresenham is a computer scientist who invented one of the most useful algorithms in computer graphics way back in 1962. The Bresenham Line Drawing Algorithm provides a very efficient way to plot a straight line between two points on a bitmap image (such as an LCD screen). The crux of the problem is illustrated in Figure 1, where we have to determine which pixels to turn on between the starting pixel (x0, y0) the finishing pixel (x1, y1).
Ideally you want to be able to do this between any arbitrary pixels, but to get a feel for this clever algorithm, let’s start with a limited example where the slope of the line is between zero (horizontal) and one (a 45 degree upward-sloping angle) and starting from the bottom left as in Figure 1. In this case it’s easy to plot the first pixel at (x0, y0), but when it comes to the second pixel, we have a decision to make.
We know the x-coordinate must be x0 + 1 but the y-coordinate could be either y0 pixels y0 + 1 (pixels A or B respectively in Figure 1). To work out which, the algorithm calculates a running error term that is the difference between the integral pixel y-coordinate and the ideal y-value given by the slope of the line. If the error is greater than 0.5, the y-value is incremented, otherwise it is not. If the y-value is incremented, then the error is adjusted to reflect the new baseline. It is easiest to see this in an example as shown in Code Fragment 1.
``````CODE FRAGMENT 1
/* Code Fragment 1 - Single octant Bresenham using floating point */
void line(int x0, int y0, int x1, int y1)
{
int x;
int y = y0;
float err = 0.0;
float derr = ((float)y1 - y0) / ((float)x1 - x0);
for(x = x0; x <= x1; x++) {
plot(x, y);
err = err + derr;
if(err > 0.5) {
y = y + 1;
err = err - 1;
}
}
}``````
This is cool but uses floating point arithmetic so it’s not very fast. Bresenham’s genius was to scale everything by 2 × dx to remove the fractional terms. This gives us the integer-only version in Code Fragment 2. This will execute much faster.
``````CODE FRAGMENT 2
/* Code Fragment 2 - Single octant Bresenham using only integer operations */
void line(int x0, int y0, int x1, int y1)
{
int x;
int y = y0;
int dx = x1 - x0;
int dy = y1 - y0;
int derr = 2 * dy - dx;
for(x = x0; x <= x1; x++) {
plot(x, y);
if(derr > 0) {
y = y + 1;
derr = derr - 2 * dx;
}
derr = derr + 2 * dy;
}
}``````
``````CODE FRAGMENT 3
/* Code Fragment 3 - Generalised Bresenham for arbitrary start and finish pixels */
void line(int x0, int y0, int x1, int y1)
{
int x, y;
int dx, dy;
int sx, sy;
int err, e2;
dx = x1 >= x0 ? x1 - x0 : x0 - x1;
dy = y1 >= y0 ? y0 - y1 : y1 - y0;
sx = x0 < x1 ? 1 : -1;
sy = y0 < y1 ? 1 : -1;
err = dx + dy;
x = x0;
y = y0;
while(1){
plot(x, y);
if((x == x1) && (y == y1)) break;
e2 = 2 * err;
if(e2 >= dy){ // step x
err += dy;
x += sx;
}
if(e2 <= dx){ // step y
err += dx;
y += sy;
}
}
}``````
But Bresenham had a bit more to offer. Code Fragment 3 (above) shows an extension to allow arbitrary start and finish pixels. Focus first on the code in the while loop. It extends the example above to select one of three possible next pixels (A, B and C in Figure 1) based on minimising a combined error term. This means it works with slopes over a whole quadrant (for example zero to 90 degrees). By setting up the initial variables based on the relative start and finish points, you effectively select any quadrant – allowing you to draw any arbitrary line. This code is neat and efficient and has been used in a graphics library I wrote many years ago and have been using ever since.
There are a number of ways these algorithms can be extended – for example, it is not much of a stretch to create a circle-drawing equivalent as shown in Code Fragment 4. This works on exactly the same principles and actually draws four quarter circles at the same time. You can’t help admiring Jack Bresenham coming up with such deceptively simple (and fast) code to draw lines circles.
``````CODE FRAGMENT 4
/* Code Fragment 4 - Circle drawing based on Bresenham */
void circle (int x0, int y0, int r)
{
int x, y;
int err, temp;
x = -r;
y = 0;
err = 2 - (2 * r);
do {
plot(x0 - x, y0 + y);
plot(x0 - y, y0 - x);
plot(x0 + x, y0 - y);
plot(x0 + y, y0 + x);
temp = err;
if(temp > x) err += ++x * 2 + 1;
if(temp <= y) err += ++y * 2 + 1;
} while (x < 0);
``````
References:
“Jack Elton Bresenham.” In Wikipedia, September 22, 2019. https://en.wikipedia.org/w/index.php?title=Jack_Elton_Bresenham&oldid=917239806
Zingl, Alois. “Rasterizing Curves,” n.d., 98. Accessed December 22, 2020 http://members.chello.at/~easyfilter/Bresenham.pdf
Keep up-to-date with our FREE Weekly Newsletter! Don't miss out on upcoming issues of Circuit Cellar. Subscribe to Circuit Cellar Magazine Note: We’ve made the Dec 2022 issue of Circuit Cellar available as a free sample issue. In it, you’ll find a rich variety of the kinds of articles and information that exemplify a typical issue of the current magazine. Would you like to write for Circuit Cellar? We are always accepting articles/posts from the technical community. Get in touch with us and let's discuss your ideas.
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Counts of human cases of West Nile
Virus by County as well as Counts of cases in horses,
dead bird counts, and sentinel mosquito samples,
sentinel chicken flocks and squirrel cases updated
regularly are available
California's West Nile Virus Website
West Nile Virus activity
in the Yuba-Sutter region has started. There are
positive mosquito pools and WNV-positive dead birds
found in both counties. There will be no more bird
pick-up for Yuba County; however, please
continue to report dead birds by calling
(1-877-968-2473) or by going
here to report online. Reporting dead birds
will help the state in mapping WNV hotspots in
Please be reminded to
continue protecting yourself from WNV by applying
mosquito repellent with DEET, avoiding the outdoors
during dusk and dawn and emptying out items with water
where mosquitoes can breed. For the latest updates regarding West Nile Virus
activity in California, please visit
California's West Nile Virus website.
Back to Top
Frequently Asked Questions (FAQs)
1. What is West
Nile Virus (WNV)?
2. How do people
and animals get West Nile Virus?
3. What are
symptoms of West Nile Virus in people?
4. How soon do
infected people get sick?
5. How can you
minimize the risk of WNV infection?
6. How is WNV
detected and monitored in California?
1. What is West Nile Virus (WNV)?
West Nile virus (WNV) is a mosquito-borne disease that
is common in Africa, west Asia, the Middle East, and
more recently, North America. Human infection with WNV
may result in serious illness. Experts believe WNV is
established as a seasonal epidemic in North America that
flares up in the summer and continues into the fall.
2. How do people and animals get West Nile Virus?
West Nile Virus is transmitted to people and animals by
infected mosquitoes. Only certain species of
mosquitoes carry the virus and very few mosquitoes are
actually infected. A mosquito first acquires the
infection by feeding on a bird with virus in its blood.
The virus lives in the mosquito and is transmitted to a
new host in the mosquito's saliva when the insect bites
a person or animal. Human-to human transmission of WNV generally does not occur. West Nile Virus can
also be transmitted via blood transfusion, transplants
or mother to child. All donated blood is checked
for WNV before being used. The risk of getting WNV
through blood transfusions and organ transplants is very
small, and should not prevent people who need surgery
from having it. Transmission during pregnancy from
mother to baby or transmission to an infant via
breastfeeding is extremely rare.
What are symptoms of West Nile Virus
people who are infected with WNV have no symptoms
whatsoever. However, of those who become ill,
symptoms can include fever, headache, nausea body aches,
mild skin rash, or swollen lymph nodes. In few
cases, the disease will progress to encephalitis
(inflammation of the brain). The time between the
mosquito bite and the onset of illness, known as the
incubation period, ranges from 5-15 days in humans.
4. How soon do infected people get sick?
People typically develop symptoms
from 3 to14 days after they are bitten by an infected
How can you minimize the risk of WNV
decrease exposure to mosquitoes and the infections they
spending time outside when mosquitoes are most active,
especially dawn and dusk
outdoors, wear long pants, long sleeve shirts and other
insect repellant, preferably with DEET, according
to label instructions
sure that doors and windows have tight fitting screens.
Repair or replace screens that have tears and holes.
all sources of standing water on your property that can
support mosquito breeding
your local mosquito and vector control agency if there
is a significant mosquito problem where you live or
is WNV detected and monitored in California?
California is well prepared to
detect, monitor, and respond to WNV through ongoing
collaboration between over 100 public agencies. The
California surveillance system includes human and horse
case detection and testing of mosquitoes, sentinel
chicken flocks, and dead birds for WNV.
Back to Top
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An international team of researchers has sequenced fragments of DNA from a pair 7,000-year-old human skeletons. In an intriguing twist, the remains, while recovered from a cave in northeastern Spain, bear little genetic resemblance to people living in the region today.
If the remains have been dated accurately, their genetic information predates that of the previous recordholder — Ötzi the Iceman — by 1,700 years, and could hold valuable clues to the link between prehistoric humans and moden European populations.
Writes LiveScience's Charles Choi:
The skeletons of two young adult males were discovered by chance in 2006 by cave explorers in a cavern high in the Cantabrian mountain range, whose main entrance is found at 4,920 feet (1,500 meters) altitude. Winters there are notably cold, which helped preserve the DNA in the bones.
These bones date back to the Mesolithic period, before agriculture spread to the Iberian Peninsula with Neolithic settlers from the Middle East. These cavemen were hunter-gatherers, judging by the ornament that one was found with of red-deer canines embroidered onto a cloth.
Paleogeneticist Carles Lalueza-Fox and his team were able to rescue the complete mitochondrial DNA from the skeleton pictured up top, which was recovered in almost perfect condition and belonged to a human the researchers have since named "Braña1." Mitochondrial DNA is the genetic information housed in sub-cellular structures called mitochondria. It is inherited only from the mother, and is incredibly useful for tracing common ancestry.
"Despite their geographical distance, individuals from the regions corresponding to the current England, Germany, Lithuania, Poland and Spain shared the same mitochondrial lineage," said Lalueza-Fox in a statement. "These hunters-gatherers shared nomadic habits and had a common origin.
The researchers went on to acquire fragments of the genetic information housed within the nuclei of the skeletons' remarkably well preserved cells. Using a technique known as shotgun sequencing, Lalueza-Fox and his colleagues were able to recover 1.34% and 0.5% of the individuals' total genomes. Their findings reveal that Braña1 and Braña2 are more closely related to contemporary populations of northern Europe than they are to modern populations of Spain and Portugal.
"Until now... the genetic affinities of the Mesolithic populations to the modern Europeans were largely unknown," write the researchers in the latest issue of Current Biology. They continue:
Our partial La Braña 1 and 2 genomic data show that modern Iberian populations are not descendants of the local hunter-gatherers inhabiting the same region prior to the arrival of farmers and thus support a genetic shift in that region between the Meso- lithic and modern populations.
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CLARKSBURG, W.Va. (WBOY) — Despite its name, none of the Ohio River is actually owned by Ohio, and it’s been quite the topic of debate since the 1700s.
Most rivers’ boundaries are split down the middle, with each state getting partial ownership, but not Ohio.
In 1783, the Ohio River was fully within the commonwealth of Virginia, which contained the land as far west as what is now Illinois and as far north as Wisconsin, including all of modern-day Ohio, West Virginia and Kentucky. But during the Confederation Congress in 1784, Virginia ceded its territory that was “to the northwest of the river Ohio” but not any of the river itself, according to the Indiana Magazine of History, making the state boundary the low-water mark on the western bank of the Ohio River.
Even though this border was made under the Articles of Confederation, it carried over under the Constitution. When Ohio became a state in 1803, it tried to take ownership of the Ohio River by claiming that the state boundary should be in the middle of the river, but the Supreme Court ruled in favor of the original owners.
When West Virginia seceded from Virginia in 1863, the ownership of the Ohio River transferred to the new state. Similarly, when the Kentucky District of Virginia became the state of Kentucky, it assumed ownership of the Ohio River.
Despite several legal battles, Ohio has not been able to claim any of the Ohio River along its border, from the northern tip of West Virginia to the Indiana border near Cincinnati.
According to the Ohio Department of Natural Resources, Ohio tried to gain ownership of the Ohio River as recently as the 1980s. In a 1966 case, the state claimed that the Ohio River in Kentucky had moved north into Ohio and said that the state line should be moved to the other side of the river, or at least to the midpoint, but the Supreme Court ruled in Kentucky’s favor. Ohio Lost another similar case in 1980.
Even now, those who are fishing in the Ohio River even from the Ohio side must follow West Virginia fishing regulations.
(Photo Courtesy: Legomine via Wikimedia Commons)
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Electricity grids that incorporate storage for power sourced from renewable resources could cut carbon dioxide emissions substantially more than systems that simply increase renewably sourced power, a new study has found.
Electrical grids that implement storage for electricity that comes from renewable resources could decrease C02 emissions more than systems which don’t store energy. The study is a first in looking at the role storage has to play in making renewable energy resources more reliable.
Power grids in California and Texas were examined in the study. Then researchers modeled which kinds of storage might make the best use of each type of renewable energy source. Then extrapolated from there how this might all affect C02 emission levels of the modeled grids.
In California it was found that a whole third of energy might not even be collected at all or simply lost from renewable sources without storage. And adding storage to this system reduced C02 emissions by 90 percent.
The study appears in Nature Communications.
The kinds of questions teachers ask children when they read books affect how much children learn, according to a new study. The study observed teachers during classroom story time and discovered the questions they ask are often too simple.
Only 24% of what teachers said when not reading the text were even questions. And those questions were answered correctly 85% of the time. While this study observed teachers, the same applies to parents and their children during story time.
Classrooms were monitored while teachers read a 25-page story called Kingdom of Friends in which two friends argue but learn to resolve their differences. All discussion was transcribed by researchers, both the teacher and children. Some five thousand questions by teachers and just under thirty five hundred child responses were recorded.
Over half, 52%, of questions were yes or no type questions. As we would expect most these questions were answered one-word style by children. The rest of the questions asked why and how.
The latter type, researchers say, are the type we need more of because they tend to produce more complex answers from the children.
The study was published by the journal called Early Childhood Research Quarterly.
Some may have noticed an unusual sight on campus at OSU this July and it, indeed, was knee-high by the forth of July. If you aren’t familiar with this colloquialism it is about corn.
A small crop of corn is growing on campus aided by soil with Com-Til; this is a compost material that uses residual biosolids from Columbus’ wastewater plants.
While it sounds a little gross, the Com-Til project is part of a long history of human’s using their own waste as an agricultural resource and is exploring what that might look like in the future. Com-Til is used all over the city to grow a variety of plants.
This is just one example of how biosolids (a nice, clean term for stuff most of us would rather not ponder) can become a resource for crop production, which in an era of rapidly increasing population and rapidly decreasing resources is a concern.
The project aims to understand what the problems and benefits of using biosolids for crop production. The project is collecting all kinds of data including the perspective of professionals and farmers in using biosolids. This will aid in one of the main goals, changing public perception of the use of such waster materials.
A research park dedicated to developing new generations of automated vehicles just opened—and OSU is part of it. The Transportation Research Center added on a 45 million dollar test facility called the Smart Mobility Advance Research Test Center (or SMARTCenter). This high tech facility is about 66% the size of Central Park in NYC.
The SMARTCenter is a collaboration between OSU, the state and JobsOhio. The connection to the test center lets the university maintain its mantel as a leader in self-driving research. Having the worlds best and newest test track in the backyard of the university will be a benefit to students and the community from an educational and job creation standpoint.
The SMARTCenter features the widest and longest data-connected test intersection in the industry. The test operation center is 10k square feet that includes research space and garages. The finished track will be an expansive test center with changeable intersections, roundabouts and road configurations.
The new test center will be fantastic opportunity for students to prepare for the jobs of the future.
It is that time of year again—the annual bicycle abatement is happening at the Ohio State University. This program, undertaken by the Dept. of Transportation and Traffic Management. The bikes are then donated to a local program.
The department estimates it collects, on average, 400-500 bikes a year. During the sweep bicycles on campus are tagged with a yellow warning notice. Bike owners have two weeks to move the bikes after tagging. Unmoved bicycles are impounded up to 90 days.
The department stated that even with hundreds of bike racks all over the campus, during the semester bike parking is always at a premium.
Even after the bikes are impounded owners have a chance to reclaim them, given they are able to prove the bike is theirs with a bike lock key, photo identification, sales receipt or some other kind of reasonable evidence of ownership.
The unclaimed bikes go to Third Hand Bike Co-op. The Columbus based nonprofit offers the repaired and safety tested bicycles at low cost to the community along with inexpensive repairs and workshops to increase ridership.
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Primitive recursive definition of the “divisibility” relation
Let $$d(x,y)= \begin{cases} 1, &\text{if }x\text{ is divisible by }y \\ 0, &\text{otherwise.} \end{cases}$$
How can I define $d(x,y)$ in terms of just the basic primitive recursive functions (zero, successor, identity, projection) and the composition and primitive recursive operations?
-
Review Link: Have you looked at common-primitive-recursive-functions? You'll find an elaborations of the basic primitive functions you refer to, and how, from those, we can obtain limited subtraction...and so forth. The link will take you to some primitive function, including division, but if you scroll to the top, and read from the start, it may shed some insight on how to define divisibility using more primitive functions as "building blocks". – amWhy Nov 11 '12 at 20:32
I just moved the link & my answer to the "comment" section below your question, and removed it from my answer (since deleted). I simply wanted you to have quick access to it if you want to revisit the post/site in the future. You just get automatically notified if someone "comments" or answers your question. – amWhy Nov 11 '12 at 22:13
The natural approach would be to define the modulus function $$m(0,y) = 0$$ $$m(x+1,y) = \begin{cases} 0 & \text{if }m(x,y)+1= y \\ m(x,y)+1 &\text{otherwise}\end{cases}$$ Then $d$ is simply $$d(x,y) = \begin{cases} 1 &\text{if }m(y,x)=0 \\ 0 &\text{otherwise}\end{cases}$$
So everything will be easy if you can express $$f(a,b,x,y) = \begin{cases} a & \text{if }x=y \\ b & \text{otherwise} \end{cases}$$ Do you have some components that might be useful for that? For example if you have the restricted subtraction function $(x,y)\mapsto \max(0,x-y)$, you can build $|x-y|$ from that, and then implement $f$ by primitive recursion over the value of $|x-y|$...
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Oh yes I do have the modified difference function you refer to, but I haven't been able to figure out how to build $d$ from this function. Can you give a hint perhaps? – Ryan Nov 11 '12 at 18:15
@Ryan: That's what I describe in the answer! – Henning Makholm Nov 11 '12 at 18:20
@Ryan: Nobody says $d$ itself needs to be defined directly by primitive recursion. In the definition I propose in my answer, it is constructed by composition: $d(x,y)=f(1,0,m(y,x),0)$. – Henning Makholm Nov 11 '12 at 18:38
@Ryan: Are you reading my answer at all? It (and my previous comment) tells you exactly what to compose. And yes, you do need define some auxiliary functions -- thougn not necessarily the modulus function if you go with Arthur's hint instead. – Henning Makholm Nov 11 '12 at 19:00
@Ryan: The part of my answer that explains how you can use absolute difference to build $f$ is this "... and then implement $f$ by primitive recursion over the value of $|x-y|$". If you're not allowed to define any auxiliary PR functions (such as modulus) during your solution simply because they are not mentioned in the text, then I think you're screwed. – Henning Makholm Nov 11 '12 at 19:12
Note, that the following functions are primitive recursive
• the (characteristic function of the) equality predicate $$\mathrm{Eq} (x,y) = \begin{cases} 1, &\text{if }x = y \\ 0, &\text{if }x \neq y; \end{cases}$$
• usual multiplication of natural numbers.
Note, also, that if $f(x,\vec{u})$ is a primitive recursive function, then so is the function $y \mapsto \sum_{x\leq y} f(x,\vec{u})$.
Using this we can define the divisibility predicate by $$d ( x , y ) = \textstyle{\sum_{z \leq x}} \mathrm{Eq} ( y \cdot z , x ).$$ (In actual fact, this equation is not that cryptic. Note that if $\chi(y,\vec{x})$ is the characteristic function of some predicate, then $\sum_{y \leq z} \chi(y,\vec{x})$ will be positive iff $\chi(y,\vec{x}) = 1$ for some $y \leq z$, and so $\min \{ 1 , \sum_{y \leq z} \chi(y,\vec{x}) \}$ will be the characteristic function for the predicate "$(\exists y \leq z) ( \chi(y,\vec{x}))$." Since given $x,y$ there is at most one $z \leq x$ such that $y \cdot z = x$, we can dispense with the "$\min$" in this specific case. The right-hand-side of the formula above can then be read as the characteristic function of the predicate "$( \exists z \leq x ) ( y \cdot z = x )$" which is exactly what we mean by "$x$ is divisible by $y$.")
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Thanks Arthur. I will save your answer for when I reach the "characteristic function" part of my course material. :) – Ryan Nov 11 '12 at 19:18
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By drilling a 1.5 mile hole deep into an Antarctic glacier, physicists working at the IceCube South Pole Observatory have captured 28 extraterrestrial neutrinos — those mysterious and extremely powerful subatomic particles that can pass straight through solid matter. Welcome to an entirely new age of astronomy.
Back in April of this year, the same team of physicists captured the highest energy neutrinos ever detected. Dubbed Bert and Ernie, the elusive subatomic particles likely originated from beyond our solar system, and possibly even our galaxy.
Neutrinos are extremely tiny and prolific subatomic particles that are born in nuclear reactions, including those that occur inside of stars. And because they're practically massless (together they contain only a tiny fraction of the mass of a single electron), they can pass through normal matter, which is why they're dubbed 'ghost particles.' Neutrinos are able to do this because they don't carry an electric charge, so they're immune to electromagnetic forces that influence charged particles like electrons and protons.
A Billion Times More Powerful
But not all neutrinos are the same. The ones discovered by the IceCube team are about a billion times more energetic than the ones coming out of our sun. A pair of them had energies above an entire petaelectron volt. That's more than 1,000 times the energy produced by protons smashed at CERN's Large Hadron Collider.
So whatever created them must have been extremely powerful. Like, mindboggingly powerful — probably the remnants of supernova explosions. Indeed, as a recent study has shown, these cosmic explosions are more powerful than we could have ever imagined — to the point where they're defying known physics.
Other candidates for neutrino production include black holes, pulsars, galactic nuclei — or even the cataclysmic merger of two black holes.
That's why the discovery of these 28 new neutrinos, and the construction of the IceCube facility, is so important. It's still a mystery, but these new findings, and the new detection technique, will help.
Back in April, the IceCube project looked for neutrinos above one petaelectronvolt, which is how Bert and Ernie were detected. But the team went back and searched through their data and found 26 neutrinos with slightly lower energies, though still above 30 teraelectronvolts that were detected between May 2010 and May 2012. While it's possible that some of these less high-energy neutrinos could have been produced by cosmic rays in the Earth's atmosphere, the researchers say that most of them likely came from space. And in fact, the data was analyzed in such a way as to exclude neutrinos that didn't come from space and other types of particles that may have tripped off the detector.
The Dawn of a New Field
"This is a landmark discovery — possibly a Nobel Prize in the making," said Alexander Kusenko, a UCLA astroparticle physicist who was not involved in the IceCube collaboration. Thanks to the remarkable IceCube facility, where neutrinos are captured in holes drilled 1.5 miles down into the Antarctic glacier, astronomers have a completely new way to scope out the cosmos. It's both literally and figuratively changing the way we see the universe.
"It really is the dawn of a new field," said Darren Grant, a University of Alberta physicist, and a member of the IceCube team.
The next phase of the project will involve the creation of a skymap — a map of the cosmos showing the directions that the neutrinos came from.
Read the entire study at Science: "Evidence for High-Energy Extraterrestrial Neutrinos at the IceCube Detector".
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The goal of segmentation is to simplify and change the representation of an image into something that is more meaningful and easier to analyze. Image segmentation is typically used to locate objects and boundaries in images. More precisely, image segmentation is the process of assigning a label to every pixel in an image such that pixels with the same label share certain visual characteristics. The result of image segmentation is a set of segments that collectively covers the entire image, or a set of contours extracted from the image. Each of the pixels in a region are similar with respect to some characteristics or computed properly, such as color, intensity or texture. Adjacent regions are significantly different with respect to same characteristics. The technique usually adopted for improving the quality of the images by segmenting the images into various regions. Many segmentation algorithms have been developed over the years. The proposed algorithm has been implemented on a set of different JPEG images. In the proposed algorithm, firstly an image is selected and then it is segmented and then the segmented image is compared with the original image.
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Third Grade Spelling and Phonics Workbook
The Third Grade Spelling and Phonics Workbook lessons are arranged in weeks. There are four pages of workbook activities per week. The fifth day of each week is a spelling test. The first day of each will feature workbook activities that will require students to write each word at least once. The main objective is for students to become familiar with the words and the spelling patterns.
Students do a variety of activities that emphasize words in and out of context and various forms of words such as with suffixes and prefixes. The spelling list still includes a phonics concept but applied to more complex words. Some lists may highlight different ways to spell the same sound or different ways similar spelling patterns are decoded.
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When we park in a big parking lot, how do we remember where we parked our car? Here's the problem facing Homer. And we're going to try to understand what's happening in his brain. So we'll start with the hippocampus, shown in yellow, which is the organ of memory. If you have damage there, like in Alzheimer's, you can't remember things including where you parked your car. It's named after Latin for "seahorse," which it resembles. And like the rest of the brain, it's made of neurons. So the human brain has about a hundred billion neurons in it. And the neurons communicate with each other by sending little pulses or spikes of electricity via connections to each other. The hippocampus is formed of two sheets of cells, which are very densely interconnected. And scientists have begun to understand how spatial memory works by recording from individual neurons in rats or mice while they forage or explore an environment looking for food. So we're going to imagine we're recording from a single neuron in the hippocampus of this rat here. And when it fires a little spike of electricity, there's going to be a red dot and a click. So what we see is that this neuron knows whenever the rat has gone into one particular place in its environment. And it signals to the rest of the brain by sending a little electrical spike. So we could show the firing rate of that neuron as a function of the animal's location. And if we record from lots of different neurons, we'll see that different neurons fire when the animal goes in different parts of its environment, like in this square box shown here. So together they form a map for the rest of the brain, telling the brain continually, "Where am I now within my environment?" Place cells are also being recorded in humans. So epilepsy patients sometimes need the electrical activity in their brain monitoring. And some of these patients played a video game where they drive around a small town. And place cells in their hippocampi would fire, become active, start sending electrical impulses whenever they drove through a particular location in that town. So how does a place cell know where the rat or person is within its environment? Well these two cells here show us that the boundaries of the environment are particularly important. So the one on the top likes to fire sort of midway between the walls of the box that their rat's in. And when you expand the box, the firing location expands. The one below likes to fire whenever there's a wall close by to the south. And if you put another wall inside the box, then the cell fires in both place wherever there's a wall to the south as the animal explores around in its box. So this predicts that sensing the distances and directions of boundaries around you — extended buildings and so on — is particularly important for the hippocampus. And indeed, on the inputs to the hippocampus, cells are found which project into the hippocampus, which do respond exactly to detecting boundaries or edges at particular distances and directions from the rat or mouse as it's exploring around. So the cell on the left, you can see, it fires whenever the animal gets near to a wall or a boundary to the east, whether it's the edge or the wall of a square box or the circular wall of the circular box or even the drop at the edge of a table, which the animals are running around. And the cell on the right there fires whenever there's a boundary to the south, whether it's the drop at the edge of the table or a wall or even the gap between two tables that are pulled apart. So that's one way in which we think place cells determine where the animal is as it's exploring around. We can also test where we think objects are, like this goal flag, in simple environments — or indeed, where your car would be. So we can have people explore an environment and see the location they have to remember. And then, if we put them back in the environment, generally they're quite good at putting a marker down where they thought that flag or their car was. But on some trials, we could change the shape and size of the environment like we did with the place cell. In that case, we can see how where they think the flag had been changes as a function of how you change the shape and size of the environment. And what you see, for example, if the flag was where that cross was in a small square environment, and then if you ask people where it was, but you've made the environment bigger, where they think the flag had been stretches out in exactly the same way that the place cell firing stretched out. It's as if you remember where the flag was by storing the pattern of firing across all of your place cells at that location, and then you can get back to that location by moving around so that you best match the current pattern of firing of your place cells with that stored pattern. That guides you back to the location that you want to remember. But we also know where we are through movement. So if we take some outbound path — perhaps we park and we wander off — we know because our own movements, which we can integrate over this path roughly what the heading direction is to go back. And place cells also get this kind of path integration input from a kind of cell called a grid cell. Now grid cells are found, again, on the inputs to the hippocampus, and they're a bit like place cells. But now as the rat explores around, each individual cell fires in a whole array of different locations which are laid out across the environment in an amazingly regular triangular grid. And if you record from several grid cells — shown here in different colors — each one has a grid-like firing pattern across the environment, and each cell's grid-like firing pattern is shifted slightly relative to the other cells. So the red one fires on this grid and the green one on this one and the blue on on this one. So together, it's as if the rat can put a virtual grid of firing locations across its environment — a bit like the latitude and longitude lines that you'd find on a map, but using triangles. And as it moves around, the electrical activity can pass from one of these cells to the next cell to keep track of where it is, so that it can use its own movements to know where it is in its environment. Do people have grid cells? Well because all of the grid-like firing patterns have the same axes of symmetry, the same orientations of grid, shown in orange here, it means that the net activity of all of the grid cells in a particular part of the brain should change according to whether we're running along these six directions or running along one of the six directions in between. So we can put people in an MRI scanner and have them do a little video game like the one I showed you and look for this signal. And indeed, you do see it in the human entorhinal cortex, which is the same part of the brain that you see grid cells in rats. So back to Homer. He's probably remembering where his car was in terms of the distances and directions to extended buildings and boundaries around the location where he parked. And that would be represented by the firing of boundary-detecting cells. He's also remembering the path he took out of the car park, which would be represented in the firing of grid cells. Now both of these kinds of cells can make the place cells fire. And he can return to the location where he parked by moving so as to find where it is that best matches the firing pattern of the place cells in his brain currently with the stored pattern where he parked his car. And that guides him back to that location irrespective of visual cues like whether his car's actually there. Maybe it's been towed. But he knows where it was, so he knows to go and get it. So beyond spatial memory, if we look for this grid-like firing pattern throughout the whole brain, we see it in a whole series of locations which are always active when we do all kinds of autobiographical memory tasks, like remembering the last time you went to a wedding, for example. So it may be that the neural mechanisms for representing the space around us are also used for generating visual imagery so that we can recreate the spatial scene, at least, of the events that have happened to us when we want to imagine them. So if this was happening, your memories could start by place cells activating each other via these dense interconnections and then reactivating boundary cells to create the spatial structure of the scene around your viewpoint. And grid cells could move this viewpoint through that space. Another kind of cell, head direction cells, which I didn't mention yet, they fire like a compass according to which way you're facing. They could define the viewing direction from which you want to generate an image for your visual imagery, so you can imagine what happened when you were at this wedding, for example. So this is just one example of a new era really in cognitive neuroscience where we're beginning to understand psychological processes like how you remember or imagine or even think in terms of the actions of the billions of individual neurons that make up our brains. Thank you very much. (Applause)
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How your brain tells you where you are
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Math problem answers are solved here step-by-step to keep the explanation clear to the students. In Math-Only-Math you'll find abundant selection of all types of math questions for all the grades with the complete step-by-step solutions.
Parents and teachers can follow math-only-math to help their students to improve and polish their knowledge. Children can practice the worksheets of all the grades and on all the topics to increase their knowledge.
Various types of Math Problem Answers are solved here.
1. Mrs. Rodger got a weekly raise of $145. If she gets paid every other week, write an integer describing how the raise will affect her paycheck. Solution: Let the 1st paycheck be x (integer). Mrs. Rodger got a weekly raise of$ 145.
So after completing the 1st week she will get $(x+145). Similarly after completing the 2nd week she will get$ (x + 145) + $145. =$ (x + 145 + 145)
= $(x + 290) So in this way end of every week her salary will increase by$ 145.
2. The value of x + x(xx) when x = 2 is:
(a) 10, (b) 16, (c) 18, (d) 36, (e) 64
Solution:
x + x(xx)
Put the value of x = 2 in the above expression we get,
2 + 2(22)
= 2 + 2(2 × 2)
= 2 + 2(4)
= 2 + 8
= 10
3. Mr. Jones sold two pipes at $1.20 each. Based on the cost, his profit one was 20% and his loss on the other was 20%. On the sale of the pipes, he: (a) broke even, (b) lost 4 cents, (c) gained 4 cents, (d) lost 10 cents, (e) gained 10 cents Solution: 20 % profit on$ 1.20
= $20/100 × 1.20 =$ 0.20 × 1.20
= $0.24 Similarly, 20 % loss on$ 1.20
= $20/100 × 1.20 =$ 0.20 × 1.20
= $0.24 Therefore, in one pipe his profit is$ 0.24 and in the other pipe his loss is $0.24. Since both profit and loss amount is same so, it’s broke even. Answer: (a) 4. The distance light travels in one year is approximately 5,870,000,000,000 miles. The distance light travels in 100 years is: (a) 587 × 108 miles, (b) 587 × 1010 miles, (c) 587 × 10-10 miles, (d) 587 × 1012 miles, (e) 587 × 10-12 miles Solution: The distance of the light travels in 100 years is: 5,870,000,000,000 × 100 miles. = 587,000,000,000,000 miles. = 587 × 1012 miles. Answer: (d) 5. A man has$ 10,000 to invest. He invests $4000 at 5 % and$ 3500 at 4 %. In order to have a yearly income of $500, he must invest the remainder at: (a) 6 % , (b) 6.1 %, (c) 6.2 %, (d) 6.3 %, (e) 6.4 % Solution: Income from$ 4000 at 5 % in one year = $4000 of 5 %. =$ 4000 × 5/100.
= $4000 × 0.05. =$ 200.
Income from $3500 at 4 % in one year =$ 3500 of 4 %.
= $3500 × 4/100. =$ 3500 × 0.04.
= $140. Total income from 4000 at 5 % and 3500 at 4 % =$ 200 + $140 =$ 340.
Remaining income amount in order to have a yearly income of $500 =$ 500 - $340. =$ 160.
Total invested amount = $4000 +$ 3500 = $7500. Remaining invest amount =$ 10000 - $7500 =$ 2500.
We know that, Interest = Principal × Rate × Time
Interest = $160, Principal =$ 2500,
Rate = r [we need to find the value of r],
Time = 1 year.
160 = 2500 × r × 1.
160 = 2500r
160/2500 = 2500r/2500 [divide both sides by 2500]
0.064 = r
r = 0.064
Change it to a percent by moving the decimal to the right two places r = 6.4 %
Therefore, he invested the remaining amount $2500 at 6.4 % in order to get$ 500 income every year.
6. Jones covered a distance of 50 miles on his first trip. On a later trip he traveled 300 miles while going three times as fast. His new time compared with the old time was:
(a) three times as much, (b) twice as much, (c) the same, (d) half as much, (e) a third as much
Solution:
Let speed of the 1st trip x miles / hr. and speed of the 2nd trip 3x / hr.
We know that
Speed = Distance/Time.
Or, Time = Distance/Speed.
So, times taken to covered a distance of 50 miles on his first trip = 50/x hr.
And times taken to covered a distance of 300 miles on his later trip = 300/3x hr.
= 100/x hr.
So we can clearly see that his new time compared with the old time was: twice as much.
7. If (0.2)x = 2 and log 2 = 0.3010, then the value of x to the nearest tenth is:
(a) -10.0, (b) -0.5, (c) -0.4, (d) -0.2, (e) 10.0
Solution:
(0.2)x = 2.
Taking log on both sides
log (0.2)x = log 2.
x log (0.2) = 0.3010, [since log 2 = 0.3010].
x log (2/10) = 0.3010.
x [log 2 - log 10] = 0.3010.
x [log 2 - 1] = 0.3010,[since log 10=1].
x [0.3010 -1] = 0.3010, [since log 2 = 0.3010].
x[-0.699] = 0.3010.
x = 0.3010/-0.699.
x = -0.4306….
x = -0.4 (nearest tenth)
8. If 102y = 25, then 10-y equals:
(a) -1/5, (b) 1/625, (c) 1/50, (d) 1/25, (e) 1/5
Solution:
102y = 25
(10y)2 = 52
10y = 5
1/10y = 1/5
10-y = 1/5
9. The fraction (5x-11)/(2x2 + x - 6) was obtained by adding the two fractions A/(x + 2) and B/(2x - 3). The values of A and B must be, respectively:
(a) 5x, -11, (b) -11, 5x, (c) -1, 3, (d) 3, -1, (e) 5, -11
Solution:
10. The sum of three numbers is 98. The ratio of the first to the second is 2/3, and the ratio of the second to the third is 5/8. The second number is:
(a) 15, (b) 20, (c) 30, (d) 32, (e) 33
Solution:
Let the three numbers be x, y and z.
Sum of the numbers is 98.
x + y + z = 98………………(i)
The ratio of the first to the second is 2/3.
x/y = 2/3.
x = 2/3 × y.
x = 2y/3.
The ratio of the second to the third is 5/8.
y/z = 5/8.
z/y = 8/5.
z = 8/5 × y.
z = 8y/5.
Put the value of x = 2y/3 and z = 8y/5 in (i).
2y/3 + y + 8y/5 = 98
49y/15 = 98.
49y = 98 × 15.
49y = 1470.
y = 1470/49.
y = 30 .
Therefore, the second number is 30.
Unsolved Questions:
1. Fahrenheit temperature F is a linear function of Celsius temperature C. The ordered pair (0, 32) is an ordered pair of this function because 0°C is equivalent to 32°F, the freezing point of water. The ordered pair (100, 212) is also an ordered pair of this function because 100°C is equivalent to 212° F, the boiling point of water.
2. A sports field is 300 feet long. Write a formula that gives the length of x sports fields in feet. Then use this formula to determine the number of sports fields in 720 feet.
3. A recipe calls for 2 1/2 cups and I want to make 1 1/2 recipes. How many cups do I need?
4. Mario answered 30% of the questions correctly. The test contained a total of 80 questions. How many questions did Mario answer correctly?
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$$\require{cancel}$$
# 3.2: Mathematics of Interference
Figure(a) shows how to determine the path length difference ΔlΔl for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle θθ between the path and a line from the slits to the screen [part (b)] is nearly the same for each path. In other words, r1r1 and r2r2 are essentially parallel. The lengths of r1r1 and r2r2 differ by ΔlΔl, as indicated by the two dashed lines in the figure. Simple trigonometry shows
Δl=dsinθ
where d is the distance between the slits. Combining this result with [link], we obtain constructive interference for a double slit when the path length difference is an integral multiple of the wavelength, or
dsinθ=mλ,form=0,±1,±2,±3,…(constructive interference).dsinθ=mλ,form=0,±1,±2,±3,…(constructive interference).
Similarly, to obtain destructive interference for a double slit, the path length difference must be a half-integral multiple of the wavelength, or
dsinθ=(m+12)λ,form=0,±1,±2,±3,…(destructive interference)dsinθ=(m+12)λ,form=0,±1,±2,±3,…(destructive interference)
where λλ is the wavelength of the light, d is the distance between slits, and θθ is the angle from the original direction of the beam as discussed above. We call m the order of the interference. For example, m=4m=4 is fourth-order interference.
(a) To reach P, the light waves from S1S1 and S2S2 must travel different distances. (b) The path difference between the two rays is ΔlΔl.
The equations for double-slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes (Figure). The closer the slits are, the more the bright fringes spread apart. We can see this by examining the equation
dsinθ=mλ,form=0,±1,±2,±3…dsinθ=mλ,form=0,±1,±2,±3…. For fixed λλ and m, the smaller d is, the larger θθ must be, since sinθ=mλ/dsinθ=mλ/d. This is consistent with our contention that wave effects are most noticeable when the object the wave encounters (here, slits a distance d apart) is small. Small d gives large θθ, hence, a large effect.
Referring back to part (a) of the figure, θθ is typically small enough that sinθ≈tanθ≈ym/Dsinθ≈tanθ≈ym/D, where ymym is the distance from the central maximum to the mth bright fringe and D is the distance between the slit and the screen. Equation may then be written as
dymD=mλ
or
ym=mλDd.
The interference pattern for a double slit has an intensity that falls off with angle. The image shows multiple bright and dark lines, or fringes, formed by light passing through a double slit.
Finding a Wavelength from an Interference PatternSuppose you pass light from a He-Ne laser through two slits separated by 0.0100 mm and find that the third bright line on a screen is formed at an angle of 10.95°10.95° relative to the incident beam. What is the wavelength of the light?
Strategy
The phenomenon is two-slit interference as illustrated in Figure and the third bright line is due to third-order constructive interference, which means that m=3m=3. We are given d=0.0100mmd=0.0100mm and θ=10.95°θ=10.95°. The wavelength can thus be found using the equation dsinθ=mλdsinθ=mλ for constructive interference.
Solution
Solving dsinθ=mλdsinθ=mλ for the wavelength λλ gives
λ=dsinθm.λ=dsinθm.
Substituting known values yields
λ=(0.0100mm)(sin10.95°)3=6.33×10−4mm=633nm.λ=(0.0100mm)(sin10.95°)3=6.33×10−4mm=633nm.
SignificanceTo three digits, this is the wavelength of light emitted by the common He-Ne laser. Not by coincidence, this red color is similar to that emitted by neon lights. More important, however, is the fact that interference patterns can be used to measure wavelength. Young did this for visible wavelengths. This analytical techinque is still widely used to measure electromagnetic spectra. For a given order, the angle for constructive interference increases with λλ, so that spectra (measurements of intensity versus wavelength) can be obtained.
Example $$\PageIndex{1}$$: Calculating the Highest Order Possible
Interference patterns do not have an infinite number of lines, since there is a limit to how big m can be. What is the highest-order constructive interference possible with the system described in the preceding example?
StrategyThe equation dsinθ=mλdsinθ=mλ (for m=0,±1,±2,±3…m=0,±1,±2,±3…) describes constructive interference from two slits. For fixed values of dandλdandλ, the larger m is, the larger sinθsinθ is. However, the maximum value that sinθsinθ can have is 1, for an angle of 90°90°. (Larger angles imply that light goes backward and does not reach the screen at all.) Let us find what value of m corresponds to this maximum diffraction angle.
SolutionSolving the equation dsinθ=mλdsinθ=mλ for m gives
m=dsinθλ.m=dsinθλ.
Taking sinθ=1sinθ=1 and substituting the values of dandλdandλ from the preceding example gives
m=(0.0100mm)(1)633nm≈15.8.m=(0.0100mm)(1)633nm≈15.8.
Therefore, the largest integer m can be is 15, or m=15m=15.
SignificanceThe number of fringes depends on the wavelength and slit separation. The number of fringes is very large for large slit separations. However, recall (see The Propagation of Light and the introduction for this chapter) that wave interference is only prominent when the wave interacts with objects that are not large compared to the wavelength. Therefore, if the slit separation and the sizes of the slits become much greater than the wavelength, the intensity pattern of light on the screen changes, so there are simply two bright lines cast by the slits, as expected, when light behaves like rays. We also note that the fringes get fainter farther away from the center. Consequently, not all 15 fringes may be observable.
Exercise $$\PageIndex{1}$$
In the system used in the preceding examples, at what angles are the first and the second bright fringes formed?
# Summary
• In double-slit diffraction, constructive interference occurs when dsinθ=mλ(form=0,±1,±2,±3…), where d is the distance between the slits, θθ is the angle relative to the incident direction, and m is the order of the interference.
• Destructive interference occurs when dsinθ=(m+12)λform=0,±1,±2,±3,…
# Conceptual Questions
Suppose you use the same double slit to perform Young’s double-slit experiment in air and then repeat the experiment in water. Do the angles to the same parts of the interference pattern get larger or smaller? Does the color of the light change? Explain.
Why is monochromatic light used in the double slit experiment? What would happen if white light were used?
# Problems
At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm?
Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm.
What is the separation between two slits for which 610-nm orange light has its first maximum at an angle of 30.0°30.0°?
Find the distance between two slits that produces the first minimum for 410-nm violet light at an angle of 45.0°.45.0°.
Calculate the wavelength of light that has its third minimum at an angle of 30.0°30.0° when falling on double slits separated by 3.00μm3.00μm. Explicitly show how you follow the steps from the Problem-Solving Strategy: Wave Optics, located at the end of the chapter.
What is the wavelength of light falling on double slits separated by 2.00μm2.00μm if the third-order maximum is at an angle of 60.0°60.0°?
At what angle is the fourth-order maximum for the situation in the preceding problem?
What is the highest-order maximum for 400-nm light falling on double slits separated by 25.0μm25.0μm?
Find the largest wavelength of light falling on double slits separated by 1.20μm1.20μm for which there is a first-order maximum. Is this in the visible part of the spectrum?
What is the smallest separation between two slits that will produce a second-order maximum for 720-nm red light?
(a) What is the smallest separation between two slits that will produce a second-order maximum for any visible light? (b) For all visible light?
(a) If the first-order maximum for monochromatic light falling on a double slit is at an angle of 10.0°10.0°, at what angle is the second-order maximum? (b) What is the angle of the first minimum? (c) What is the highest-order maximum possible here?
Shown below is a double slit located a distance x from a screen, with the distance from the center of the screen given by y. When the distance d between the slits is relatively large, numerous bright spots appear, called fringes. Show that, for small angles (where sinθ≈θsinθ≈θ, with θθ in radians), the distance between fringes is given by Δy=xλ/dΔy=xλ/d
Using the result of the preceding problem, (a) calculate the distance between fringes for 633-nm light falling on double slits separated by 0.0800 mm, located 3.00 m from a screen. (b) What would be the distance between fringes if the entire apparatus were submersed in water, whose index of refraction is 1.33?
Using the result of the problem two problems prior, find the wavelength of light that produces fringes 7.50 mm apart on a screen 2.00 m from double slits separated by 0.120 mm.
In a double-slit experiment, the fifth maximum is 2.8 cm from the central maximum on a screen that is 1.5 m away from the slits. If the slits are 0.15 mm apart, what is the wavelength of the light being used?
The source in Young’s experiment emits at two wavelengths. On the viewing screen, the fourth maximum for one wavelength is located at the same spot as the fifth maximum for the other wavelength. What is the ratio of the two wavelengths?
If 500-nm and 650-nm light illuminates two slits that are separated by 0.50 mm, how far apart are the second-order maxima for these two wavelengths on a screen 2.0 m away?
Red light of wavelength of 700 nm falls on a double slit separated by 400 nm. (a) At what angle is the first-order maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
## Glossary
fringes
bright and dark patterns of interference
order
integer m used in the equations for constructive and destructive interference for a double slit
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||1 - 3
||1 hour with cemetery visit or 30 minutes without
||Classroom and cemetery (optional)
||The students will be able to compare burial practices between the Woodland Culture and our culture.
crayons or markers
**optional--cemetery to visit
||cemetery burial mound
- Discuss with the students that they will be visiting the Effigy Mounds National Monument and explain to them that they will be seeing a cemetery for a culture of people that lived thousands of years ago. We call this culture of people the Woodland Culture.
- Ask the students: How do we bury people that have died? Have a discussion and affirm all answers. Be sure to include the following points about our burial customs:
- we put the person in a casket
- we have a funeral service
- we dig a hole about 6 feet deep, put the casket in the ground and pile the dirt back in the hole until it is even with the ground around it
- we place a headstone where the person is buried, telling who the person was
- Visit a local cemetery (some children may never have been to one). Walk through the cemetery so the children get a feel for what the cemetery looks and feels like. While at the cemetery, have the children draw what they see with crayons
- Discuss how the people of the Woodland Culture buried their dead. Be sure to include all 4 types of burials. See Background information and outline on mound styles.
- Have the children picture what this cemetery may look like and have them draw a picture of it.
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As we walked through the autumn forest and noticed the falling leaves, I asked the kids what happened to the piles of leaves we walked through last year. That started a conversation about decomposition.
Decomposition and Composting
Decomposition is the process in which organic material is broken down into simpler forms of matter (according to Wikipedia). It’s natural recycling!
When leaves fall and plants and animals die, they start this process of breaking down or decay. Insects, bacteria, and fungus all carry out decomposition. In the end, dead matter decays and is turned back into soil. That’s what happened to the piles of leaves from last fall.
You might have a compost bin or pile at your house where your yard, garden, and kitchen waste are decomposed to create nutrient rich soil that can be put back into the garden.
Make Your Own Compost Cups
To study composting up close, we decided to create our own mini compost bins in cups, so we could see decomposition in action. You can make your own compost cups science project with these easy steps.
- 16 ounce plastic cup
- Organic items such as grass clippings, kitchen scraps (no meat or dairy), leaves, coffee grinds, bark, etc.
- Plastic wrap
- 1 tablespoon water
- 1/4 cup dirt
- Rubber band or tape
1. Place organic material, dirt, and water in the plastic cup.
2. Cover the cup with plastic wrap and seal with a rubber band or tape. Give it a good shake and place it in a warm, sunny place like a window or safe spot outside where it won’t be disturbed.
3. Every couple of days add another tablespoon of water and give it a shake. Note what is happening to the organic matter.
The warm environment of the cup increases the activity of the microbes inside. These bacteria and fungus go to work breaking down the organic matter in the cup. The added water and oxygen from the shaking keep the process going.
Within a day or two you can see this happening. Given enough time, you’ll be able to see the organic matter turn into dark, nutrient rich compost that can be added to garden soil.
By contributing writer Marci
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Thinking Mathematically by Robert Blitzer
Document Sample
``` Thinking
Mathematically
Chapter 11:
Counting Methods and
Probability Theory
Thinking
Mathematically
Section 1:
The Fundamental Counting
Principle
The Fundamental Counting Principle
If you can choose one item from a group of
M items and a second item from a group of
N items, then the total number of two-item
choices is M N.
You MULTIPLY the numbers!
The Fundamental Counting Principle
At breakfast, you can have eggs, pancakes or cereal.
You get a free juice with your meal: either OJ or apple
juice. How many different breakfasts are possible?
eggs pancakes cereal
OJ apple OJ apple OJ apple
1 2 3 4 5 6
Example: Applying the Fundamental
Counting Principle
• The Greasy Spoon Restaurant offers 6
appetizers and 14 main courses. How many
different meals can be created by selecting
one appetizer and one main course?
• Using the fundamental counting principle,
there are 14 6 = 84 different ways a
person can order a two-course meal.
Example: Applying the Fundamental
Counting Principle
• This is the semester that you decide to take your
required psychology and social science courses.
• Because you decide to register early, there are 15
sections of psychology from which you can
choose. Furthermore, there are 9 sections of social
science that are available at times that do not
conflict with those for psychology. In how many
ways can you create two-course schedules that
satisfy the psychology-social science requirement?
Solution
The number of ways that you can satisfy the
requirement is found by multiplying the
number of choices for each course.
You can choose your psychology course
from 15 sections and your social science
course from 9 sections. For both courses
you have:
15 9, or 135 choices.
The Fundamental Counting
Principle
The number of ways a series of successive
things can occur is found by multiplying the
number of ways in which each thing can
occur.
Example: Options in Planning a
Course Schedule
Next semester you are planning to take three
courses - math, English, and humanities. Based
on time blocks and highly recommended
professors, there are 8 sections of math, 5 of
English, and 4 of humanities that you find
suitable. Assuming no scheduling conflicts, there
are:
8 5 4 = 160 different three course schedules.
Example
Car manufacturers are now experimenting with
lightweight three-wheeled cars, designed for a
driver and one passenger, and considered ideal for
city driving. Suppose you could order such a car
with a choice of 9 possible colors, with or without
air-conditioning, with or without a removable
roof, and with or without an onboard computer. In
how many ways can this car be ordered in terms of
options?
Solution
This situation involves making choices with
four groups of items.
color - air-conditioning - removable roof - computer
9 2 2 2 = 72
Thus the car can be ordered in 72 different
ways.
Example: A Multiple Choice Test
You are taking a multiple-choice test that
has ten questions. Each of the questions has
four choices, with one correct choice per
question. If you select one of these options
per question and leave nothing blank, in
how many ways can you answer the
questions?
Solution
We DON’T blindly multiply the first two numbers
we see. The answer is not 10 4 = 40.
We use the Fundamental Counting Principle to
determine the number of ways you can answer the
test. Multiply the number of choices, 4, for each of
the ten questions
4444444444
=1,048,576
Example: Telephone Numbers in
the United States
Telephone numbers in the United States
begin with three-digit area codes followed
by seven-digit local telephone numbers.
Area codes and local telephone numbers
cannot begin with 0 or 1. How many
different telephone numbers are possible?
Solution
We use the Fundamental Counting Principle
to determine the number of different
telephone numbers that are possible.
8 10 10 8 10 10 10 10 10 10
=6,400,000,000
Thinking
Mathematically
Section 2:
Permutations
Permutations
• A permutation is an arrangement of
objects.
– No item is used more than once.
– The order of arrangement makes a difference.
Example: Counting
Permutations
Based on their long-standing contribution to
rock music, you decide that the Rolling
Stones should be the last group to perform
at the four-group Offspring, Pink Floyd,
Sublime, Rolling Stones concert. Given
this decision, in how many ways can you
put together the concert?
Solution
We use the Fundamental Counting Principle to
find the number of ways you can put together the
concert. Multiply the choices:
3211=6
3 choices 2 choices 1 choice 1 choice
offspring whichever only one
pink floyd of the two remaining
stones
sublime remaining
Thus, there are six different ways to arrange the
concert if the Rolling Stones are the final group to
perform.
Example: Counting
Permutations
You need to arrange seven of your favorite
books along a small shelf. How many
different ways can you arrange the books,
assuming that the order of the books makes
a difference to you?
Solution
You may choose any of the seven books for the
first position on the shelf. This leaves six choices
for second position. After the first two positions
are filled, there are five books to choose from for
the third position, four choices left for the fourth
position, three choices left for the fifth position,
then two choices for the sixth position, and only
one choice left for the last position.
7 6 5 4 3 2 1 = 5040
There are 5040 different possible permutations.
Factorial Notation
If n is a positive integer, the notation n! is
the product of all positive integers from n
down through 1.
n! = n(n-1)(n-2)…(3)(2)(1)
note that 0!, by definition, is 1.
0!=1
Permutations of n Things Taken r at a
Time
The number of permutations possible if r
items are taken from n items:
n!
nPr = (n – r)! = n(n – 1) (n – 2) (n – 3) . . . (n – r + 1)
n! = n(n – 1) (n – 2) (n – 3) . . . (n – r + 1) (n - r) (n - r - 1) . . . (2)(1)
(n – r)! = (n - r) (n - r - 1) . . . (2)(1)
Permutations of n Things Taken r at a
Time
The number of permutations possible if
r items are taken from n items:
nPr: starting at n, write down r numbers
going down by one:
nPr = n(n – 1) (n – 2) (n – 3) . . . (n – r + 1)
1 2 3 4 r
Problem
A math club has eight members, and it must choose 5
officers --- president, vice-president, secretary, treasurer
and student government representative. Assuming that
each office is to be held by one person and no person can
hold more than one office, in how many ways can those
five positions be filled?
We are arranging 5 out of 8 people into the five distinct
offices. Any of the eight can be president. Once selected,
any of the remaining seven can be vice-president.
Clearly this is an arrangement, or permutation, problem.
8P5 = 8!/(8-5)! = 8!/3! = 8 · 7 · 6 · 5 · 4 = 6720
Permutations with duplicates.
• In how many ways can you arrange the
letters of the word minty?
• That's 5 letters that have to be arranged, so
the answer is 5P5 = 5! = 120
• But how many ways can you arrange the
letters of the word messes?
• You would think 6!, but you'd be wrong!
messes
here are six permutations of messes
me s s e s 1
well, all 3! arrangements of the s's
me s s e s 2 look the same to me!!!!
me s s e s This is true for any arrangement
3
of the six letters in messes, so
me s s e s 4 every six permutations should
count only once.
me s s e s 5 The same applies for the 2!
arrangement of the e's
6
Permutations with duplicates.
• How many ways can you arrange the letters
of the word messes?
• The problem is that there are three s's and 2
e's. It doesn't matter in which order the s's
are placed, because they all look the same!
• This is called permutations with duplicates.
Permutations with duplicates.
• Since there are 3! = 6 ways to arrange the
s's, there are 6 permutations that should
count as one. Same with the e's. There are
2! = 2 permutations of them that should
count as 1.
• So we divide 6! by 3! and also by 2!
• There are 6!/3!2! = 720/12 = 60 ways to
arrange the word messes.
Permutations with duplicates.
• In general if we want to arrange n items, of which
m1, m2, .... are identical, the number of
permutations is
n!
m1!m2!m3!
Problem
A signal can be formed by running different
colored flags up a pole, one above the other.
Find the number of different signals
consisting of 6 flags that can be made if 3
of the flags are white, 2 are red, and 1 is
blue
6!/3!2!1! = 720/(6)(2)(1) = 720/12 = 60
Thinking
Mathematically
Section 3:
Combinations
Combination: definition
A combination of items occurs when:
• The item are selected from the same
group.
• No item is used more than once.
• The order of the items makes no
difference.
How to know when the problem is a
permutation problem or a
combination problem
• Permutation:
– arrangement, arrange
– order matters
• Combination
– selection, select
– order does not matter.
Example: Distinguishing between
Permutations and Combinations
• For each of the following problems, explain if the
problem is one involving permutations or
combinations.
• Six students are running for student government
president, vice-president, and treasurer. The
student with the greatest number of votes becomes
the president, the second biggest vote-getter
becomes vice-president, and the student who gets
the third largest number of votes will be student
government treasurer. How many different
outcomes are possible for these three positions?
Solution
• Students are choosing three student
government officers from six candidates.
The order in which the officers are chosen
makes a difference because each of the
offices (president, vice-president, treasurer)
is different. Order matters. This is a
problem involving permutations.
Example: Distinguishing between
Permutations and Combinations
• Six people are on the volunteer board of
supervisors for your neighborhood park. A
three-person committee is needed to study
the possibility of expanding the park. How
many different committees could be formed
from the six people on the board of
supervisors?
Solution
• A three-person committee is to be formed
from the six-person board of supervisors.
The order in which the three people are
selected does not matter because they are
not filling different roles on the committee.
Because order makes no difference, this is a
problem involving combinations.
Example: Distinguishing between
Permutations and Combinations
• Baskin-Robbins offers 31 different flavors
of ice cream. One of their items is a bowl
consisting of three scoops of ice cream,
each a different flavor. How many such
bowls are possible?
Solution
• A three-scoop bowl of three different flavors is to
be formed from Baskin-Robbin’s 31 flavors. The
order in which the three scoops of ice cream are
put into the bowl is irrelevant. A bowl with
chocolate, vanilla, and strawberry is exactly the
same as a bowl with vanilla, strawberry, and
chocolate. Different orderings do not change
things, and so this problem is combinations.
Combinations of n Things Taken r at a
Time
n n!
= nCr =
r r!(n – r)!
Note that the sum of the two numbers on the bottom
(denominator) should add up to the number on the
top (numerator).
Computing Combinations
• Suppose we need to compute 9C3
9! 9!
9 C3
9
3!(9 3)! 3!6!
3!6!
• r = 3, n – r = 6
• The denominator is the factorial of smaller of
the two: 3!
Computing Combinations
• Suppose we need to compute 9C3
9! 9!
9 C3
9
3!(9 3)! 3!6!
3!6!
• r = 3, n – r = 6
• In the numerator write (the product of) all the
numbers from 9 down to n - r + 1 = 6 + 1 = 7:
• There should be the same number of terms in
the numerator and denominator: 9 8 7
Computing Combinations
• If called upon, there's a fairly easy way to
compute combinations.
– Given nCr , decide which is bigger: r or n – r.
– Take the smaller of the two and write out the
factorial (of the number you picked) as a
product.
– Draw a line over the expression you just wrote.
Computing Combinations
• If called upon, there's a fairly easy way to
compute combinations.
– Now, put n directly above the line and directly
above the leftmost number below.
– Eliminate common factors in the numerator and
denominator.
– Do the remaining multiplications.
– You're done!
Computing Combinations
• Suppose we need to compute 9C3 .
– n – r = 6, and the smaller of 3 and 6 is 3.
3 4
987
= 3 4 7 = 84
321
1 1
Finding Probabilities from
Odds
• If the odds in favor of an event E are a to b,
then the probability of the event is given by
P( E ) a
ab
Finding Probabilities from
Odds
• If the odds against an event E are a to b,
then the probability of the event is given
by
P( E ) b
ab
Finding Probabilities from
Odds
• Example:
– Suppose Bluebell is listed as 7:1 in the
– The odds listed on a horse are odds
against that horse winning, that is, losing.
– The probability of him losing is
7 / (7+1) = 7/8.
– The probability of him winning is 1/8.
Finding Probabilities from
Odds
• Example:
– Suppose Bluebell is listed as 7:1 in the third
race at the Meadowlands. (a:b against)
– The odds listed on a horse are odds against that
horse winning, that is, losing.
– The probability of him losing is
a
7 / (7+1) = 7/8. a b
b
– The probability of him winning is 1/8. a b
ab
Thinking
Mathematically
Section 7:
Events Involving And;
Conditional Probability
Independent Events
• Two events are independent events if the
occurrence of either of them has no effect
on the probability of the other.
• For example, if you roll a pair of dice two
times, then the two events are independent.
What gets rolled on the second throw is not
affected by what happened on the first
throw.
And Probabilities with
Independent Events
• If A and B are independent events, then
P(A and B) = P(A) P(B)
• The example of choosing from four pairs of
socks and then choosing from three pairs of
shoes (= 12 possible combinations) is an
example of two independent events.
Dependent Events
• Two events are dependent events if the occurrence
of one of them does have an effect on the
probability of the other.
• Selecting two Kings from a deck of cards by
selecting one card, putting it aside, and then
selecting a second card, is an example of two
dependent events.
• The probability of picking a King on the second
selection changes because the deck now contains
only 51, not 52, cards.
And Probabilities with
Dependent Events
• If A and B are dependent events, then
• P(A and B) =
P(A) P(B given that A has occurred)
• written as
P(A) P(B|A)
Conditional Probability
• The conditional probability of B, given A,
written P(B|A), is the probability that event
B will occur computed on the assumption
that event A has occurred.
• Notice that when the two events are
independent, P(B|A) = P(B).
Conditional Probability
• Example:
– Suppose you are picking two cards from a deck
of cards. What is the probability you will pick a
King and then another face card?
– The probability of an King is 4 =
1 .
52 13
– Once the King is selected, there are 11 face cards
left in a deck holding 51 cards.
– P(A) = 1 . P(B|A) = 11
13 51
– The probability in question is 1 11
13 51
Applying Conditional
Probability to Real-World Data
P(B|A) =
observed number of times B and A occur together
observed number of times A occurs
Review
P(not E) 1 – P(E)
P(A or B) P(A) + P(B) mutually
– P(A and B) exclusive:
P(A) + P(B)
P(A and B) P(A) P(B|A) independent:
P(A) P(B)
Odds in favor - P(E) / P(not E) probability is
a:b a/(a+b)
Odds against - P(not E) / P(E) probability is
a:b b/(a+b)
Thinking
Mathematically
Section 8:
Expected Value
Expected Value
• Expected value is a mathematical way to use
probabilities to determine what to expect in various
situations over the long run.
• For example, we can use expected value to find the
outcomes of the roll of a fair dice.
• The outcomes are 1, 2, 3, 4, 5, and 6, each with a
probability of 1 . The expected value, E, is computed
6
by multiplying each outcome by its probability and
• E = 1 1 + 2 1 + 3 1 + 4 1 + 5 1 + 6 1
6 6 6 6 6 6
= (1+2+3+4+5+6)/6 = 21 = 3.5
6
Expected Value
E = 1 1 + 2 1 + 3 1 + 4 1 + 5 1 + 6 1
6 6 6 6 6 6
= (1 + 2 + 3 + 4 + 5 + 6)/6 = 21 = 3.5
6
Of course, you can't roll a 3½ . But the
average value of a roll of a die over a long
period of time will be around 3½.
Example Expected Value and Roulette
A roulette wheel has 38 different
"numbers."
• One way to bet in roulette is to place \$1
on a single number.
• If the ball lands on that number, you are
awarded \$35 and get to keep the \$1 that
you paid to play the game.
• If the ball lands on any one of the other
37 slots, you are awarded nothing and
the \$1 you bet is collected.
Example Expected Value and Roulette
• 38 different numbers.
• If the ball lands on your number,
you win awarded \$35 and you keep
the \$1 you paid to play the game.
• If the ball lands on any of the other
37 slots, you are awarded nothing
and you lose the \$1 you bet.
• Find the expected value for playing roulette if
you bet \$1 on number 11 every time. Describe
what this means.
Solution
Outcome Gain/Loss Probability
1
11 \$35 38
\$1
37
Not 11 38
E = \$35( 1 ) + (-\$1)( 37)
38 38
35 37 2
= \$ - \$ = -\$ ≈ -\$0.05
38 38 38
This means that in the long run, a player can
expect to lose about 5 cents for each game
played.
Expected Value
• A real estate agent is selling a house. She gets a 4-
month listing. There are 3 possibilities:
– she sells the house: (30% chance) earns \$25,000
– another agent sells
the house: (20% chance) earns \$10,000
– house not sold: (50% chance) loses \$5,000
• What is the expected profit (or loss)?
• If the expected profit is at least \$6000 she would
consider it a good deal.
Expected Value
Outcome Probability Profit or product
loss
she sells 0.3 +\$25,000 +\$7,500
other sells 0.2 +\$10,000 +\$2,000
doesn't sell 0.5 -\$5,000 -\$2,500
+\$7,000
The realtor can expect to make \$7,000.
Make the deal!!!!
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Spindly trees, rusted gates, crumbling stone, a solitary mourner— these things come to mind when we think of cemeteries. But not so long ago, many burial grounds were lively places, with blooming gardens and crowds of people strolling among the headstones. How did our cemeteries become what they are today? Some have been around for centuries, like the world’s largest, Wadi al-Salaam, where more than five million people are buried. But most of the places we’d recognize as cemeteries are much younger. In fact, for much of human history, we didn’t bury our dead at all. Our ancient ancestors had many other ways of parting with the dead loved ones. Some were left in caves, others in trees or on mountaintops. Still others were sunk in lakes, put out to sea, ritually cannibalized, or cremated. All of these practices, though some may seem strange today, were ways of venerating the dead. By contrast, the first known burials about 120,000 years ago were likely reserved for transgressors, excluding them from the usual rites intended to honor the dead. But the first burials revealed some advantages over other practices: they protected bodies from scavengers and the elements, while shielding loved ones from the sight of decay. These benefits may have shifted ancient people’s thinking toward graves designed to honor the dead, and burial became more common. Sometimes, these graves contained practical or ritual objects, suggesting belief in an afterlife where the dead might need such tools. Communal burials first appeared in North Africa and West Asia around 10 to 15,000 years ago, around the same time as the first permanent settlements in these areas. These burial grounds created permanent places to commemorate the dead. The nomadic Scythians littered the steppes with grave mounds known as kurgans. The Etruscans built expansive necropoles, their grid-patterned streets lined with tombs. In Rome, subterranean catacombs housed both cremation urns and intact remains. The word cemetery, or “sleeping chamber,” was first used by ancient Greeks, who built tombs in graveyards at the edges of their cities. In medieval European cities, Christian churchyards provided rare, open spaces that accommodated the dead, but also hosted markets, fairs, and other events. Farmers even grazed cattle in them, believing graveyard grass made for sweeter milk. As cities grew during the industrial revolution, large suburban cemeteries replaced smaller urban churchyards. Cemeteries like the 110-acre Père-Lachaise in Paris or the 72-acre Mt. Auburn in Cambridge, Massachusetts were lushly landscaped gardens filled with sculpted stones and ornate tombs. Once a luxury reserved for the rich and powerful, individually marked graves became available to the middle and working classes. People visited cemeteries for funerals, but also for anniversaries, holidays, or simply an afternoon outdoors. By the late 19th century, as more public parks and botanical gardens appeared, cemeteries began to lose visitors. Today, many old cemeteries are lonely places. Some are luring visitors back with tours, concerts, and other attractions. But even as we revive old cemeteries, we’re rethinking the future of burial. Cities like London, New York, and Hong Kong are running out of burial space. Even in places where space isn’t so tight, cemeteries permanently occupy land that can’t be otherwise cultivated or developed. Traditional burial consumes materials like metal, stone, and concrete, and can pollute soil and groundwater with toxic chemicals. With increasing awareness of the environmental costs, people are seeking alternatives. Many are turning to cremation and related practices. Along with these more conventional practices, people can now have their remains shot into space, used to fertilize a tree, or made into jewelry, fireworks, and even tattoo ink. In the future, options like these may replace burial completely. Cemeteries may be our most familiar monuments to the departed, but they’re just one step in our ever-evolving process of remembering and honoring the dead.
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The fascinating history of cemeteries
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NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.3 are part of NCERT Solutions for Class 8 Maths. Here we have given NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.3.
Board CBSE Textbook NCERT Class Class 8 Subject Maths Chapter Chapter 3 Chapter Name Understanding Quadrilaterals Exercise Ex 3.3 Number of Questions Solved 12 Category NCERT Solutions
## NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.3
Question 1.
Given a parallelogram ABCD. Complete each statement along with the definition with the definition or property used.
(ii) ∠DCB = …………….
(iii) OC = ……………….
(iv) m∠DAB + m∠CDA = …………..
Solution.
Opposite sides of a parallelogram are equal
(ii) ∠DCB = ∠DAB
Opposite angles of a parallelogram are equal
(iii) OC = OA
∵ Diagonals of a parallelogram bisect each other
(iv) m∠DAB + m∠CDA = 180°
∵ Adjacent angles of a parallelogram are supplementary.
Question 2.
Consider the following parallelo¬grams. Find the values of the unknowns x, y, z.
Solution.
(i) y = 100°
Opposite angles of a parallelogram are equal
x + 100° = 180°
Adjacent angles in a parallelogram are supplementary
⇒ x = 180° – 100°
⇒ x = 80°
⇒ z – x = 80°
Opposite angles of a parallelogram are of equal measure
(ii) x + 50° = 180°
Adjacent angles in a parallelogram are supplementary
⇒ x = 180° – 50° = 130°
⇒ y = x = 130°
The opposite angles of a parallelogram are of equal measure
180° – z = 50°
Opposite angles of a parallelogram are of equal measure
⇒ z = 180° – 50° = 130°
(iii) x = 90°
Vertically opposite angles are equal
x + y + 30° = 180°
By angle sum property of a triangle
⇒ 90° + y + 30° = 180°
⇒ 120° + y = 180°
⇒ y = 180° – 120° = 60° z + 30° + 90° – 180°
By angle sum property of a triangle
z = 60°
(iv) y = 80°
Opposite angles of a parallelogram are of equal measure
x + 80° = 180°
Adjacent angles in a parallelogram are supplementary
⇒ x = 180° – 80°
⇒ x = 100°
⇒ 180°-2+ 80°= 180°
Linear pair property and adjacent angles in a parallelogram are supplementary.
z = 80°
(v) y = 112°
Opposite angles of a parallelogram are equal
x + y + 40° = 180°
By angle sum property of a triangle
⇒ x + 112° + 40° = 180°
⇒ x + 152° = 180°
⇒ x = 180°- 152°
⇒ x = 28°
z = x = 28°.
Alternate interior angles
Question 3.
Can a quadrilateral ABCD be a parallelogram if
(i) ∠D + ∠B = 180° ?
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm
(iii) ∠A = 70° and ∠C = 65°?
Solution.
(i) Can be, but need not be
(ii) No: in a parallelogram, opposite sides are equal; but here, AD ≠ BC.
(iii) No: in a parallelogram, opposite angles are of equal measure; but here ∠A ≠ ∠C.
Question 4.
Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
Solution.
A kite, for example
Question 5.
The measures of two adjacent angles of a parallelogram are in the ratio 3:2. Find the measure of each of the angles of the parallelogram.
Solution.
Let the two adjacent angles be 3x° and 2x°.
Then,
3x° + 2x° = 180°
∴ Sum of the two adjacent angles of a parallelogram is 180°
⇒ 5x° = 180°
⇒ $${ x }^{ \circ }=\frac { { 180 }^{ \circ } }{ 5 }$$
⇒ x° = 36°
⇒ 3x° = 3 x 36° = 108°
and
2x° = 2 x 36° = 72°.
Since, the opposite angles of a parallelogram are of equal measure, therefore the measures of the angles of the parallelogram are 72°, 108°, 72°, and 108°.
Question 6.
Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Solution.
Let the two adjacent angles of a parallelogram be x° each.
Then,
x° + x° = 180°
∴ Sum of the two adjacent angles of a parallelogram is 180°.
⇒ 2x° = 180°
⇒ $${ x }^{ \circ }=\frac { { 180 }^{ \circ } }{ 2 }$$
⇒ x° = 90°.
Since the opposite angles of a parallelogram are of equal measure, therefore the measure of each of the angles of the parallelogram is 90°, i.e., each angle of the parallelogram is a right angle.
Question 7.
The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
Solution.
x = 180° – 70° = 110°
Linear pair property and the opposite angles of a parallelogram are of equal measure.
∵ HOPE is a || gm
∴ HE || OP
and HP is a transversal
∴ y = 40°
alternate interior angles
40° + z + x = 180°
The adjacent angles in a parallelogram are supplementary
⇒ 40° + z + 110° = 180°
⇒ z + 150° = 180°
⇒ z = 180° – 150°
⇒ z = 30°.
Question 8.
The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)
(i)
(ii)
Solution.
(i)
For Figure GUNS
Since the opposite sides of a parallelogram are of equal length, therefore,
⇒ 3x = 18
⇒ $$x=\frac { 18 }{ 3 } =6$$
and, 3y – 1 = 26
⇒ 3y = 26 + 1
⇒ 3y = 27
$$y=\frac { 27 }{ 3 } =9$$
Hence, x = 6; y = 9.
(ii)
For Figure RUNS
Since the diagonals of a parallelogram bisect each other, therefore,
⇒ x + y = 16 …(1)
and, y + 7 = 20 …(2)
From (2),
⇒ y – 20 – 7 = 13
Putting y = 13 in (1), we get
⇒ x + 13 = 16 ⇒ x = 16 – 13 = 3.
Hence, x = 3; y = 13.
Question 9.
In the below figure both RISK and CLUE are parallelograms. Find the value of x.
Solution.
Question 10.
Explain how this figure is a trapezium. Which of its two sides is parallel?
Solution.
∵ ∠KLM + ∠NML = 80° + 100° = 180°
∴ KL || NM
∵ The sum of consecutive interior angles is 180°
∴ Figure KLMN is a trapezium.
Its two sides $$\overline { KL }$$ and $$\overline { NM }$$ are parallel.
Question 11.
Find m∠C in the figure, if $$\overline { AB }$$ || $$\overline { DC }$$.
Solution.
∵ $$\overline { AB }$$ || $$\overline { DC }$$
∴ m∠C + m∠B = 180°
∵ The sum of consecutive interior angles is 180°
m∠C+ 120° = 180°
⇒ m∠C = 180° – 120° = 60°.
Question 12.
Find the measure of ∠P and ∠S, if $$\overline { SP }$$ || $$\overline { RQ }$$ in the figure. (If you find mZ R, is there more than one method to find m∠P ?)
Solution.
∵ $$\overline { SP }$$ || $$\overline { RQ }$$
∴ m∠P+m∠Q = 180°
∵ The sum of consecutive interior angles is 180°
⇒ m∠P + 130° = 180°
⇒ m∠P = 180° – 130°
⇒ m∠P = 50°
Again, m∠R + m∠S = 180°
∵ The sum of consecutive interior angles is 180°
⇒ 90° + m Z S = 180°
⇒ m∠S = 180° – 90° = 90°
Yes; there is one more method of finding m∠P if m∠R is given and that is by using the angle sum property of a quadrilateral.
We have,
m∠P + m∠Q + m∠R + m∠S = 360°
⇒ m∠P + 130° + 90° + 90° = 360°
⇒ m∠P + 310° = 360°
⇒ m∠P = 360° – 310° = 50°.
We hope the NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.3 help you. If you have any query regarding NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.3, drop a comment below and we will get back to you at the earliest.
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If you have ever used a navigation service to find optimal route and estimate time to destination, you've used algorithms on graphs. Graphs arise in various real-world situations as there are road networks, computer networks and, most recently, social networks! If you're looking for the fastest time to get to work, cheapest way to connect a set of computers into a network or efficient algorithm to automatically find communities and opinion leaders in Facebook, you're going to work with graphs and algorithms on graphs. In this online course, you will first learn what a graph is and what are some of the most important properties. Then you'll learn several ways to traverse graphs and how you can do useful things while traversing the graph in some order. We will then talk about shortest paths algorithms — from the basic ones to those which open door for 1000000 times faster algorithms used in Google Maps and other navigational services. You will use these algorithms if you choose to work on our Fast Shortest Routes industrial capstone project. We will finish with minimum spanning trees which are used to plan road, telephone and computer networks and also find applications in clustering and approximate algorithms.
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PITTSBURGH—Scientists at Carnegie Mellon University have discovered that our ears use the most efficient way to process the sounds we hear, from babbling brooks to wailing babies. These results represent a significant advance in understanding how sound is encoded for transmission to the brain, according to the authors, whose work is published with an accompanying "News and Views" editorial in the Feb. 23 issue of Nature.
The research provides a new mathematical framework for understanding sound processing and suggests that our hearing is highly optimized in terms of signal coding—the process by which sounds are translated into information by our brains—for the range of sounds we experience. The same work also has far-reaching, long-term technological implications, such as providing a predictive model to vastly improve signal processing for better-quality compressed digital audio files and designing brain-like codes for cochlear implants, which restore hearing to the deaf.
To achieve their results, the researchers took a radically different approach to analyzing how the brain processes sound signals. Abstracting from the neural code at the auditory nerve, they represented sound as a discrete set of time points, or a "spike code," in which acoustic components are represented only in terms of their temporal relationship with each other. That's because the intensity and basic frequency of a given feature are essentially "kernalized," or compressed mathematically, into a single spike. This is similar to a player piano roll that can reproduce any song by recording what note to press when the spike code encodes any natural sound in terms of the precise timings of the elemental acoustic features. Remarkably, when the researchers derived the optimal set of features for natural sounds, they corresponded exactly to the patterns observed by neurophysiologists in the auditory nerves.
"We've found that timing of just a sparse number of spikes actually encodes the whole range of nature sounds, including components of speech such as vowels and consonants, and natural environment sounds like footsteps in a forest or a flowing stream," said Michael Lewicki, associate professor of computer science at Carnegie Mellon and a member of the Center for the Neural Basis of Cognition (CNBC). "We found that the optimal code for natural sounds is the same as that for speech. Oddly enough, cats share our own optimal auditory code for the English language."
"Our work is the only research to date that efficiently processes auditory code as kernalized spikes," said Evan Smith, a graduate student in psychology at the CNBC. Until now, scientists and engineers have relied on Fourier transformations—initially discovered 200 years ago—to separate and reconstitute parameters like frequency and intensity as part of traditional sound signal processing. "Our new signal processing framework appears far more efficient, effective and concise in conveying a rich variety of natural sounds than anything else," Lewicki said.
Smith and Lewicki's approach dissects sound based only on the timing of compressed "spikes" associated with vowels (like cat vocalizations), consonants (like rocks hitting one another) and sibilants (ambient noise).
To gather sounds for their research, the scientists traipsed through the woods and recorded cracking branches, crunching leaves and wind rustling through leaves before returning to the laboratory to decode the information contained in this rich set of sounds. They also discovered what they consider the most "natural" sound: if they play back a random set of spikes, it sounds like running water.
"We're very excited about this work because we can give a simple theoretical account of the auditory code which predicts how we could optimize signal processing to one day allow for much more efficient data storage on everything from DVDs to iPods," Lewicki said.
"For instance, if we could use a cochlear implant to 'talk' to the auditory nerve in a more natural way via our discovered coding, then we could quite possibly design implants that would convey sounds to the brain that are much more intelligible," he said.
The authors' research, which combines computer science, psychology, neuroscience and mathematics, is funded by the National Institutes of Health and the National Science Foundation.
The CNBC is dedicated to understanding the neural mechanisms that give rise to cognitive processes, including learning and memory, language and thought, perception and attention, and planning and action. The CNBC faculty includes researchers with primary and joint appointments in the departments of Biological Sciences, Computer Science, Psychology, Robotics and Statistics at Carnegie Mellon; and Bioengineering, Mathematics, Neurobiology, Neurology, Neuroscience, Psychiatry and Psychology at the University of Pittsburgh. See http://www.cnbc.cmu.edu for more information.
Byron Spice | 412-268-9068 | bspice [atsymbol] cs.cmu.edu
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Now have the students compare columns H and I. These are the totals from the two questionnaires given. Should the scores be the same? Why would you think that they should be the same? If they are not the same, give a reason/reasons why they might not be the same. They may respond with several different answers. Two main points should be addressed. These points are (1) either the test questions are not asking the same thing when translated from English to Spanish or (2) perhaps the participants did not understand the questionnaire in one of the languages. The next lesson will focus on analyzing this question: Was the questionnaire translated accurately?
Spreadsheet, HHIE-S, computer skill, data analysis, cross cultural adaptation, hearing handicap inventory elderly
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"url": "http://teachhealthk-12.uthscsa.edu/activity/activity-2c-comparing-spanish-and-english-hhie-s-scores"
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