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Introduction: Oscilloscope Music
This Instructable is to fulfill a requirement for the documentation portion of the microcomputer interfacing project at Utah State University.
Step 1: Background
An oscilloscope is used to display and measure a voltage signal that is plotted against time. An oscilloscope in XY mode plots a signal against another signal sort of like a parametric equation. This project uses an oscilloscope in XY mode to display images produced by a sound file.
Step 2: Original Idea
The original idea for the project was to convert an old Cathode Ray Tube (CRT) television set into an XY oscilloscope and use that to display the images. This can be done by disconnecting the deflection coils. When you disconnect the horizontal coils a vertical line appears, and when you disconnect the vertical coil, a horizontal line appears. All I had to do was connect the audio source to the deflection coils and I would have an XY oscilloscope. Unfortunately, I ran into several problems.
Step 3: Problems Encountered
One of the problems I encountered was the safety features. The TV was able to detect that it’s deflection coils had been disconnected and would not turn on. This is to prevent the electron beam from burning a hole in the phosphor on the screen. I measured the resistance of the coils and placed a resistor across it. The resistor immediately burned in half because of the high voltages. I tried again using a higher rated resistor, but that didn’t work either. I read some forums online about how another set of deflection coils could be hooked up to the original TV, so I found another TV and hooked up it’s deflection coil to mine. The impedance wasn’t the same so it didn’t turn on. After some more research I found that older TVs did not have the safety feature and didn’t care if it’s deflection coils were disconnected. I was able to find a TV produced in 2000 that seemed to work. I was able to get some simple shapes on the screen, but anything more complicated than a circle would be heavily distorted. Eventually this TV stopped working and it kept blowing fuses.
I was able to find a small TV that was made in 1994. This TV worked pretty well, but I wasn’t able to get the correct orientation of the image, even when I switched the signals in every combination. It also had the same problems as the other TV and would not produce complicated images. After a lot of research I found out that problem was that I was trying to produce a vector image on a raster display. A raster display is a screen that scans horizontally very quickly and then vertically at a slower rate. A vector display uses lines to produce images. I found tutorials on how to convert a raster display to a vector display, but the process was dangerous and would take a long time.
Step 4: Solution
After all of these problems, I was able to find a pretty simple solution; an XY oscilloscope emulator program that took audio as an input. Once I found this program, I switched from focusing on creating an oscilloscope to creating a way to produce an audio file from an image to display on an oscilloscope.
Step 5: Edge Detection and Matlab Program
Here is a basic flowchart of my program. It starts out with an image that is loaded into the EdgeDetect.m MATLAB program. This program converts it to a gray-scale image and then detects the edges in the image. The XY coordinates of the detected edges are placed into two arrays which are converted into a sound file.
Step 6: Example: Instructables Robot
Here is an example of the process with the instructables robot. First download an image of the instructables robot and save it as "image.png" into your MATLAB working folder (same place as "EdgeDetect.m"). Make sure the image doesn't have anything you want to be detected or it could add a bunch of unnecessary coordinates into your sound file. Run the EdgeDetect program and the image will be converted to gray-scale, and have it's edges detected and stored as a sound file named "vector.wav". Next open up the sound file in Audacity or another sound editing program. Open up your oscilloscope emulator program (link in previous step), set the sample rate to 192000 Hz, press start, click the microphone button, and select the line in option. In Audacity press "shift + spacebar" to play the sound file in a loop. The image should appear on the oscilloscope emulator.
Step 7: Troubleshooting/Example Files
As I developed this program I had to adjust some settings in the program. Here are some things to double check if it is not working:
-Make sure your audio output is being fed into your line in on your computer and that you have 2 separate (left & right) audio channels
-If the image is not being read by the MATLAB program you may need to edit it in paint and save it as a different format.
-On line 61 of the code, be sure to include the numbers from the edge detect screen. The program usually puts a rectangle around the whole thing which you can cut out by changing it from "i=1:length(B)" to "i=2:length(B)". Also, if you have specific numbers that you want to include, but don't want to include them all, you can use square brackets to get specific numbers: "[ 1 3 6 10 15 17]"
-If the image looks shaky and the parts are all over the place you may need to reduce the number of samples by adjusting "N" on line 76. The simpler the image the lower N can be, but it should be higher if the image is complex. For the robot I used N=5.
-You can also adjust "Fs" on line 86. The higher the sampling rate the better the image will look, but some sound cards will not be able to handle higher sampling rates. Modern songs have a sampling rate around 320000 Hz. | <urn:uuid:c60fe4e9-3327-4768-9c47-52adb71fd7cd> | {
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Density of Solids and Liquids
Lesson 2 of 11
Objective: Students will be able to measure the density of solids (regular and irregular shapes) and liquids.
By 8th grade, students should have already been introduced to measuring volume, mass and density of solids and liquids. This lesson serves to review and reinforce these concepts. This lab does not include measuring the density of gases. That can be found in another of my lessons.
I include this lesson here even though it is not an explicitly addressed in the NGSS Middle School standards as there are still many students in grades 6-8 who will not have mastered measuring density of matter.
This lesson should be able to completed in one 50 minute class period. A list of materials needed is included in the resources.
Place one set of materials on each tray. If using salt water, make sure to prepare this ahead of time and have in 30 ml beakers.
While this lab does not involve any dangerous chemicals, I always encourage teachers to have students get into the habit of using safety goggles with all labs.
Begin with an elicit activity.
Ask students to open their science journals and respond to the following questions:
What is mass? What is volume? What is density?
Give them 5 minutes to write down their ideas then turn and talk to their table group, sharing their responses. I use this time to listen and monitor group discussions. Should you notice that certain students are not speaking, ask them what their thoughts are in order to pull them into the discussion.
Ask each table group to come up with a working definition that they all agree on for mass, volume and density. The goal here is not a textbook definition but a working definition that makes sense to your students.
To help those students who might have limited English language skills or need more support, consider providing a scaffold. Perhaps a visual image or partial definition along with images to help them put these ideas together.
Bring the whole group back together and ask each table to share their working definitions.
Guide the class into drafting a working definition for each term and record this on chart paper or on a word wall.
If using science journals with indexes, add the words to the index at this time if you like.
Show students a marble, a die and a screw and ask them how they would determine the mass and volume for each. Most of them will now to use a scale for the mass but measuring the volume may produce some varied responses.
If your students have not learned about calculating the volume of a geometric solid, this is good place to discuss the formulas used to measure the volumes of spheres, cubes and cylinders. Here is a lesson you may consider from fellow BetterLesson Master Teacher Erin Doughty on Making Sense of the Formula for determining volume.
For the screw, since it is an irregularly shaped solid, another method is needed, water immersion. Commonly refereed to as Archimedes' Principle, where the volume of the submerged portion equals the volume of fluid it displaces. This is great place for a demonstration. You will need a 100 ml graduated cylinder filled with 50 ml of water and an irregularly shaped object that will fit down into the cylinder.
Here is a YouTube video on water displacement to show your students. You may want to turn off the volume -- there is no narration but there is music. It would be important to pause the video as it pays to check on student understanding.
Note: if you are using a glass cylinder, you should teach about measuring the volume by finding the meniscus, however many of the Nalgene® polymethylpentene graduated cylinders nowadays do not have a meniscus.
With the students reading along, review the Density Lab Student Sheet to be sure everyone understands the expectations of the lab. I like read through the introduction and directions as a class to help ensure that my students understand what they are being asked to do and why. I teach this lesson at the beginning of the year and have found that the more time I take to instill good habits, like reading the lab first, helps prevent procedural errors later.
With your lab trays all set up and your students all having read the lab and asked their clarifying questions, it is time to start the lab.
While the students are working on the lab, I like to circulate around the class to check in with each group to be sure they are on track and help any students who might need additional assistance.
I find it to be a good practice to have students share their data to the class. You might create a data table on the whiteboard, poster paper or project it from a laptop then have students come and fill in their data as they finish each section of the lab. Once complete, I like to discuss the data and what patterns we see.
Assuming everyone measured the same or similar objects, is the data consistent from group to group or not and why? When determining the density of these objects, can we determine an average? How would we do that?
In the student work below you can see where their average densities for the same objects are all different. By collecting and averaging a larger sample we get averages closer to the actual densities of the metals.
Students need to practice the skill of analyzing data and discussing both in small group and as a class. While not as glamorous as the lab's themselves, data analysis is a key skill that students must develop and use when making claims based on evidence. The analysis will aid them in valuing the importance of accuracy and precision of data as they learn the rigor required to be consistent when collecting data during labs and investigations, a skill that is still very much in development at the middle grades. | <urn:uuid:79c7bbce-6bba-438c-8a01-c3ac090d896f> | {
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1) to investigate learning of novel words over multiple trials and over time in children with ASD relative to peers
2) to determine whether words are more easily learned in social versus linguistic contexts
3) to determine how children with ASD utilize social or linguistic cues to word learning using eye-tracking techniques
Methods: Our participants included 13 children with ASD, 13 children with language delay and 13 typically developing children (aged seven years). Children saw three novel objects on a computer screen and clicked the photo that matched a spoken sentence. In the social cue condition, a female gazed at the target item. In the linguistic cue condition, information in the sentence biased a particular interpretation. We recorded children’s eye-movements as they completed the task. Immediately after the experiment and approximately four weeks later, we assessed word learning via word recognition, definition and naming tasks.
Results: In the recognition task, all participants identified more words learned with social cues than linguistic cues. Similarly, all groups provided more semantic information in definitions for items presented in the social condition, even though semantic information was explicitly stated in the linguistic condition. In the naming task, there was an interaction between group and cue type such that participants with ASD were better at recalling phonological information for words presented with social versus linguistic cues, whereas type of cue did not affect performance in the comparison groups. We are currently analysing eye-tracking data, focusing on the hypothesis that children with ASD are able to devote more processing resources to phonological information in the social cues condition because they do not spend as much time studying the social cue (i.e. the face) as much as peers.
Conclusions: Our results indicate that social cues such as eye gaze and head turn are particularly salient cues for word learning, even for ASD participants. More semantic information was recalled in the social cue condition, suggesting the possibility that social cues are mapped quickly, leaving more time to encode visual features of novel objects. The most notable finding of this study is that children with ASD were better than peers at phonological aspects of word learning, especially when words were presented with social cues. Our eye-tracking analyses will enable us to determine whether success on this task is the result of devoting more attention to sound than meaning. We consider these findings as an alternative mechanism for acquiring vocabulary in ASD. | <urn:uuid:828a2229-d028-4158-b887-52fd53ca686b> | {
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At the very heart of the Milky Way is a region known as Sagittarius A. This region is known the be the home of a supermassive black hole with millions of times the mass of our own Sun. And with the discovery of this object, astronomers have turned up evidence that there are supermassive black holes at the centers most most spiral and elliptical galaxies.
The best observations of Sagittarius A*, using Very Long Baseline Interferometry radio astronomy have determined that it’s approximately 44 million km across (that’s just the distance of Mercury to the Sun). Astronomers have estimated that it contains 4.31 million solar masses.
Of course, astronomers haven’t actually seen the supermassive black hole itself. Instead, they have observed the motion of stars in the vicinity of Sagittarius A*. After 10 years of observations, astronomers detected the motion of a star that came within 17 light-hours distance from the supermassive black hole; that’s only 3 times the distance from the Sun to Pluto. Only a compact object with the mass of millions of stars would be able to make a high mass object like a star move in that trajectory.
The discovery of a supermassive black hole at the heart of the Milky Way helped astronomers puzzle out a different mystery: quasars. These are objects that shine with the brightness of millions of stars. We now know that quasars come from the radiation generated by the disks of material surrounding actively feeding supermassive black holes. Our own black hole is quiet today, but it could have been active in the past, and might be active again in the future.
Some astronomers have suggested other objects that could have the same density and gravity to explain Sagittarius A, but anything would quickly collapse down into a supermassive black hole within the lifetime of the Milky Way.
We have written many articles about Sagittarius A. Here’s an article about how the Milky Way’s black hole is sending out flares, and even more conclusive evidence after 16 years of observations.
We have recorded an episode of Astronomy Cast all about the Milky Way. Give it a listen: Episode: 99 – The Milky Way | <urn:uuid:1155550e-23aa-436f-9366-f2de5aff9a4f> | {
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Battle of Trenton
The German Hessian troops had occupied Trenton since December 14, 1776, while Gen. George Washington and the Continental Army encamped on the Pennsylvania side of the Delaware River, following their retreat across New Jersey beginning in Fort Lee on November 20. On the morning after the famous Christmas night crossing of the Delaware River by Gen. Washington and his troops, the Americans surprised the Hessian Troops.
Gen. Washington's plan was to arrive in Trenton before dawn, under the cover of darkness, so as to surprise the Hessians. However, they had difficulties crossing the Delaware River through the snow and arrived late. Fortunately for Gen. Washington's army, the surprise had been maintained. The Hessians were caught off guard by the attack and defeated decisively.
The American victory cost only several American casualties but inflicted substantial casualties to the Hessians: 22 dead, 83 wounded, and approximately 900 taken as prisoners of war. Among the Hessian casualties was their Commanding Officer, Col. Johann Gottlieb Rall.
The victory at the first Battle of Trenton turned around the face of the war. After months of defeat and retreat for the American Army, this victory changed the morale of both the army and the country. It was followed up over the next ten days by additional victories at the Second Battle of Trenton and the Battle of Princeton.
Today, see the Trenton Battle Monument staue at the site where the American army was positioned, as it was the high ground and offered an excellent position for the cannons to fire down on the Hessian army. Stop by the Old Barracks Museum, which provides tours and interpretations of American colonial life with artifacts and weapons, and a gift shop.
Every December the commemorate these battles, the Trenton Downtown Association hosts Patriots Week held during the week between Christmas and New Year's. It attracts thousands of visitors full of art, music, literature, and living history events. The week ends with the Battle of Trenton Reenactment, where you follow the troops through the streets and watch as they relive the events of these fateful battles. | <urn:uuid:f27829b7-b081-4e7a-8c01-b7aa6c561ba6> | {
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# Calculus-Applied Optimization Problem:
posted by .
Find the point on the line 6x + 3y-3 =0 which is closest to the point (3,1).
Note: Your answer should be a point in the xy-plane, and as such will be of the form (x-coordinate,y-coordinate)
• Calculus-Applied Optimization Problem: -
what a stupid note. Of course it will be a point with (x,y) coordinates! This is a calculus class - you know all that already.
Given a point (x,y) on the line, y=1-2x
The distance from (x,1-2x) to (3,1) is
d^2 = (x-3)^2 + (1-(1-2x))^2 = 5x^2-6x+9
so, we want minimum d, when dd/dx = 0:
2d dd/dx = 10x-6
dd/dx = 10x-6/2sqrt(blah blah)
dd/dx=0 when x = 3/5.
So, minimum d^2 is
5(3/5)^2 - 6(3/5) + 9 = 36/5
minimum d is 6/√5
Or, as we all know, the distance from a point (x,y) to a line ax+by+c=0 is
|ax+by+c|/√(a^2+b^2) = (6(3)+3(1)-3)/√(36+9) = 18/√45 = 6/√5 | crawl-data/CC-MAIN-2017-26/segments/1498128320841.35/warc/CC-MAIN-20170626170406-20170626190406-00059.warc.gz | null |
# Thread: Counting Problem Using Product rule and Subtraction rule (plz check my answer)
1. ## Counting Problem Using Product rule and Subtraction rule (plz check my answer)
Problem: There are 10 people in a line, where each person is either male or female. How many different lineups are there, where there are either 5 consecutive men, or 6 consecutive women?
Here's my answer:
For the case of 5 consecutive men: say you have the lineup M M M M M _ _ _ _ _ then for the other 5 spaces to the right you can either choose a man or a woman. Using the product rule, there are 2^5 different ways to create a lineup with 5 consecutive men in the first 5 spots. Now if you shift the 5 consecutive men 6 times to the right, you get a total of 6 * 2^5 different ways to create a lineup with 5 consecutive men.
For the case of 5 consecutive women: This would be the same situation as above and we would be able to create a total of 6 * 2^5 different ways to create a lineup with 5 consecutive women.
But there are 2 cases where there are both 5 consecutive men and 5 consecutive women such as: M M M M M W W W W W and W W W W W M M M M M
So the answer for the amount of different lineups is 12*2^5 - 2
is my answer correct?
2. ## Re: Counting Problem Using Product rule and Subtraction rule (plz check my answer)
Hey HeartyBowl.
If you have five consecutive men (or women) then the next person must be the opposite gender. In light of this you have to slightly adjust your analyses. You will then have to shift this and take into account that the tail (as opposed to the head of the line) just before (and after) must also be the opposite gender.
See if you can take the above into account to get a new answer.
Also if you meant to say "at least" so many people then disregard my response.
3. ## Re: Counting Problem Using Product rule and Subtraction rule (plz check my answer)
I get what you're saying and now that i realize it, the problem isn't being clear enough. I don't know if it means "at least" or if it means "at most".... | crawl-data/CC-MAIN-2018-26/segments/1529267864139.22/warc/CC-MAIN-20180621094633-20180621114633-00056.warc.gz | null |
The laws of supply and demand are in play in any market, wherever people are buying and selling goods and services. The labor market is no different. While we talk about the labor market as if were one monolithic market, within the overall labor force of 155 million, there are many subsections, each subject to the laws of supply and demand.
The supply of labor is determined by population, immigration and labor force participation -- how many adults are working or actively seeking employment. The supply of labor can be influenced by additional workers entering the labor force, which tends to depress wage rates. Workers leaving the labor force, either because they are retiring or becoming discouraged about finding a job, tend to support wage rates. The higher the wage rate, the more people want to work, but the supply side is just half the market.
The demand for labor derives from the demand for the goods and services that labor produces. A strong demand for products creates a demand for the labor to produce them. When the wage rate is high, employers limit the number of employees they hire. Workers who improve their skills can improve the demand for their services, since they are more productive to their employers.
Every time the Congress votes to increase the minimum wage, a debate erupts about its effect on unemployment. If workers are employed at the prevailing minimum wage, then a government-mandated higher minimum will increase unemployment. More workers will look for work seeking the higher wage; at the same time, employers will lay off workers because they cannot -- or will not -- pay them more than the free market dictates.
Unions typically negotiate to raise their pay scale or to limit the size of their membership. A craft union composed of members in a particular trade can restrict the supply of labor by requiring that employers hire only union workers. With a limited supply of workers, the result is higher wages. An inclusive union organizes all available workers and then engages in collective bargaining for higher wages. Higher wages reduce the demand for workers being hired.
The effect of inexpensive labor from abroad has depressed U.S. wages even as jobs have been lost. The Third World has a virtually unlimited supply of labor, which is becoming more skilled with the passage of time. American labor must gain new skills to remain competitive in a global market.
- Hemera Technologies/AbleStock.com/Getty Images | <urn:uuid:0131fa82-4451-487b-8386-8cf233838aca> | {
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Books open windows to the world. They have the ability to transform readers, expand knowledge and support learning both in and out of the classroom. The Great American Read (TGAR) provides a variety of ways to increase engagement around books, and develop students’ literacy and language skills.
Here are some innovative ways to deepen student learning around classic novels. You can choose a novel from The Great American Read, or venture out and select one your own.
Encourage your students, children or teens to write a short story, essay or poem about a book they loved, and why. Below are some ideas to get your started!
Ask students to discuss the following questions with a classmate, and summarize their answers in a written statement:
What is one of your favorite books and why? Who is your favorite character and what makes him/her so special to you?
Ask students to write a Haiku (a 3 sentence poem) about one of their favorite books:
Haiku Example: Adventures of Huckleberry Finn by Mark Twain
Teen runs from dad
Befriends a runaway slave
Satire of Old South
Write a blog about a book you are reading, including details about characters (Ex: smart, rambunctious, witty, intense), the setting, plots or themes.
Encourage students to read a book from The Great American Read. Ask them to write letters back and forth between characters from the book.
Encourage your students, children or teens to express their creative side through art, music and media activities. Below are some ideas to get you started!
Ask students to design a book cover combining themes or characters from two different novels on The Great American Read list. Show your Great American Read spirit, and share the project online using #GreatReadPBS | <urn:uuid:13654b30-10e6-4472-a981-ade71555a8ac> | {
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Technology has boomed over the last 100 years. We know more and can do more than ever before. As Nikola Tesla said, “The day science begins to study non-physical phenomena, it will make more progress in one decade than in all the previous centuries of existence”. From breakthroughs in aeronautics and aerospace to mobile technology and the internet, we have advanced an awful lot, awfully quickly. We are able to produce an exact map of the stars, set to any date since 1900.
Let’s not get too complacent, however. Not only is there so much more for us to discover, it’s easy to write off the past as a web of primitive cultures, feeling their way through life with only mythical tales to explain the world around them. This couldn’t be further from the truth and there are many technologies which remain completely unexplained, even with the help of modern science. Lest we also forget that without the giants which came before us there we wouldn’t have the leap in understanding which we have today. Let’s take a look at 10 ancient technologies you won’t believe could have existed!
1 - Greek Fire
Greek fire was not invented by the Ancient Greeks, but the Byzantine Greeks in the 7th century CE and it was an invention imported from the Middle East. It could be used both on land and at sea to devastating effect. The exact composition of Greek fire is unknown, but it is likely to include naphtha and quicklime, as it would ignite when it made contact with water. Terrifyingly, sailors were unable to put out the flames as water did not help.
Greek fire could be sprayed, under pressure, to launch flames at enemy ships and, later, it was used on lands to attack fortifications or as a defensive weapon. Napalm is the modern successor of this deadly, but incredible, feat of chemistry.
2 - The Antikythera Mechanism
The Antikythera mechanism is an ancient Greek computer, used to predict astrological movements, as well as determine the dates and times of various games, including the Olympic Games. It was discovered in 1901 among the wreckage of a trading ship and has been dated to around 100 BCE.
The Antikythera mechanism is the first known example using scientific dials and used 30 gear wheels. The level of complexity behind this geared mechanism was not matched for a further millennium – until medieval cathedral clocks were built. The drive train for tracking the Moon’s position is extremely sophisticated, using epicyclic gearing and a slot-and-pin mechanism to mirror the variations in the Moon’s movement across the sky.
3 - The Great Pyramids of Giza
The Great Pyramid of Giza is one of the Seven Wonders of the Ancient World. It is one of three huge pyramids on a large plain, west of Cairo, built around 4500 years ago. From the time of Herodotus to the modern day, people have speculated at how they were built. The Egyptians of the time are believed to have had only rudimentary tools, they were unfamiliar with the wheel, they had no machinery such as cranes, a limited knowledge of astronomy and only copper tools. It seems that it was achieved with brute force – a huge amount of labourers who would have taken around 20 years to complete each pyramid.
It is not just the size of the structures which is mind-blowing, but the precision with which it was built. The pyramids are oriented to within 1/15th of a degree to north, south, east and west. The Great Pyramid of Giza is the most accurately aligned structure in existence (including all our modern buildings). It is also located at the centre of the earth’s land mass. The east/west parallel and north/south meridian intersect perfectly with the Great Pyramid.
The mathematics behind the pyramids’ constructions are also incredible. The radius of the sun is related to the perimeter of the granite coffers and the weight of the pyramid is relative to the earth’s mass. This seemingly impossible construction has led many to believe that the Egyptians must have had alien technology to help them.
4 - Non-Rusting Iron Pillar of Delhi
Delhi is home to an iron pillar, over one thousand years old, which has never rusted. Not until in the invention of Stainless Steel by Harry Brearley in 1913 was it possible to prevent rusting, yet it was achieved in the reign of King Chandra. The iron pillar is 7.2 metres high with a 16-inch diameter. It weighs over 3,000 kg and has ancient writing on it, which has been preserved due to the lack of rusting.
After careful analysis, scientists found the pillar had undergone a three-phase process which created a thin protective over the pillar. This surface layer quickly oxidised but protected the iron pillar underneath from rusting. Like so much ancient knowledge, the method for preventing iron from rusting was lost for hundreds of years after.
5 - The Baghdad Battery
The Baghdad Battery, as it has come to be called, is a set of three objects: a tube of copper, a rod of iron and a ceramic pot. Discovered in Iraq near Ctesiphon, it is believed to date from either the Sasanian or Parthian empires of Persia, anywhere from 150 BCE to 650 AD.
The purpose of the Baghdad Battery is unknown, but the most famous theory is that it worked as a battery for either electroplating or electrotherapy. Corrosion in the metal supports the idea that it may have had an acidic electrolyte solution added to create electricity. The current most prevalent theory amongst academics, however, suggests this was simply a storage vessel for sacred scrolls. Even if they didn’t quite cross the finish line to creating electricity, they weren’t far off!
6 - Archimedes’ Heat-Ray Weapons
Another technology whose existence is debated is Archimedes’ heat-ray weapon. There are ancient writings, however, which reference this ancient invention. It is said that during the Siege of Syracuse (in which Archimedes was killed), large metal mirrors (of either copper or bronze) were used to direct the Sun’s rays onto enemy ships, setting them on fire.
A Greek scientist named Ioannis Sakkas recreated the heat-ray weapons to test whether it would have been possible. He trained 70 mirrors coated with copper onto a mock roman warship from around 50 metres away. The boat burned to a cinder within seconds. The real boats would have also had tar on them which would have aided in the combustion. Whether Archimedes did, in fact, create a heat-ray weapon is unknown, but the ingenuity of the Greek mathematician, physicist, inventor, engineer and astronomer is not in doubt.
7 - Viking Compass
The Viking compass, also known as the runic compass or the Vegvisir, was made of eight Viking rune staves. It was a symbol of protection and is also believed to have been used as a compass with the same accuracy as a modern-day GPS. The Vikings used sunstones to navigate at sea, and the Viking compass is thought to be a smaller version of these.
The eight rune staves could correspond to the cardinal and intercardinal directions (North, North East, East, South East etc.). With a nail placed in the centre, its shadow would reveal the directions. Today, the Viking compass acts as a symbol of spiritual guidance for the Asatru faith and symbolizes the Icelandic culture.
8 - Stonehenge
Archaeologists believe Stonehenge was constructed between 3000 BCE and 2000 BCE and it is a UNESCO World Heritage Site and British icon. It took 75 giant stones to construct Stonehenge. How exactly the huge stones of Stonehenge were transported and put in place is still the subject of debate, as there is virtually no evidence of construction techniques at that time. It is theorized that the stones were moved, either rolled across logs or on a sleigh with animal fat greasing the track beneath.
Transporting and placing the stones is not the only impressive thing about this megalithic creation. Where the stones were placed betrays another level of ancient knowledge. Stonehenge has a celestial observatory function, which may well have been used to predict eclipses, solstices, equinoxes and more. People from all around the world come to celebrate the summer solstice and winter equinox at Stonehenge. This incredible construction can be seen as an ancient precursor to our modern technology allowing us to track and create maps of the stars.
9 - Ancient Model Aircraft
Artefacts belonging to the ancient Egyptian and Central American cultures look astonishingly similar to modern-day aircraft. The Egyptian artefact was found in a tomb in Saqqara in 1898. The six-inch wooden object appears to have a fuselage, a tail, wings, and even a pilot’s seat. The object is believed to be aerodynamic enough to glide. This one-thousand-year-old object, made of gold, could be mistaken for a replica of a delta wing aircraft.
10 - Rocks of Sacsayhuamán
Sacsayhuamán is a megalithic site, predating the Incas. It is made of stones weighing over 50-tonnes cut so precisely and assembled to such a degree of perfection, as though the rocks must have melted into place, that modern engineers are over-awed by the sight of it. Not even a single sheet of paper could slide between the rocks!
Sacsayhuamán stands 3701 metres above sea level in the outskirts of Cusco, Peru. Parts of the structure have bent corners which seem to imply they were able to somehow soften the stone. Similar signs of workmanship beyond their means has been found in Egypt. Perhaps the Inca legends are true and Sacsayhuamán was built by the gods?
These are just a handful of some of the wonderous and inexplicable technologies of the ancient world. They are the first giant leaps in mankind’s understanding of the world and our capabilities within it. Though many of these technologies became buried beneath the sands of time, it is important to remember that it was the ingenuity of civilizations past, with relatively little to work with, to create the stepping stones which have led us to where we are today, and where we will go tomorrow. Our ability to create a map of the stars, from any time over the past 120 years, would not be possible without men like Archimedes or our instinctual gravitation towards the celestial realms, as in Stonehenge or the Great Pyramids of Giza. | <urn:uuid:29ee8f07-c705-4f8f-8b13-6e360ce162cd> | {
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# Verbal Reasoning - Data Sufficiency - Discussion
Discussion Forum : Data Sufficiency - Section 1 (Q.No. 1)
Directions to Solve
In each of the questions below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and
Give answer
• (A) If the data in statement I alone are sufficient to answer the question, while the data in statement II alone are not sufficient to answer the question
• (B) If the data in statement II alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question
• (C) If the data either in statement I alone or in statement II alone are sufficient to answer the question
• (D) If the data given in both statements I and II together are not sufficient to answer the question and
• (E) If the data in both statements I and II together are necessary to answer the question.
1.
Question: In which year was Rahul born ?
Statements:
1. Rahul at present is 25 years younger to his mother.
2. Rahul's brother, who was born in 1964, is 35 years younger to his mother.
I alone is sufficient while II alone is not sufficient
II alone is sufficient while I alone is not sufficient
Either I or II is sufficient
Neither I nor II is sufficient
Both I and II are sufficient
Answer: Option
Explanation:
From both I and II, we find that Rahul is (35 - 25) = 10 years older than his brother, who was born in 1964. So, Rahul was born in 1954.
Discussion:
22 comments Page 2 of 3.
Mht said: 10 years ago
Mother age is not given, so we can not get mother age by option 1.
Asif said: 10 years ago
We don't consider the answer but judge whether the statements are sufficient or not.
Gouthami said: 10 years ago
In this the answer is not important to find the year we use single conclusion or double statement is important and above problem clearly understood that Rahul 10 years elder to his brother so to conclude both statements are necessary to answer the question.
Yogeshwar Goswami said: 1 decade ago
Hey Guys I will help you.
Suppose Rahul's mother age is 50.
So Rahul is 25.
And his brother is 15.
So brother was in born 1964.
So Rahul is older than brother.
That's why his birth age is less than brother.
Nagesha said: 1 decade ago
Rahul is elder to his Brother and He was born in 1954. And In statement no where Rahul age is mentioned. The answer i.e E, that we need both the statement to answer this question.
Hasha said: 1 decade ago
It should be 1954 Rahul is elder son to his mother, when his brother is born in 1964 so obviously he should born earlier i.e. 1954.
Jagdeesh said: 1 decade ago
The answer is 1974 because rahul is the elder one and he should have been born in 1974.
Satyendra gangwar said: 1 decade ago
The answer is 1974 because Rahul brother age 35.
And his brother age 25 year old.
Mahendra said: 1 decade ago
I think answer is 2nd option, why because with help of 1st option we can't get answer.
Layeeq said: 1 decade ago
25 years younger to his mother, but not 25 years old.
Post your comments here:
Your comments will be displayed after verification. | crawl-data/CC-MAIN-2024-33/segments/1722641311225.98/warc/CC-MAIN-20240815173031-20240815203031-00208.warc.gz | null |
# x^2 = 2^x. What is x ??
Yes, this is inspired by a homework question, the answer to which I obtained via graphing calculator. But it leads me to wonder how you would actually solve this equation. I started with taking logarithms of both sides but that only muddled things up more. One of the three answers is obvious (2; seeing as 2^2 = 2^2) and the others are x=4 and x= approx. -.767 ; how can you obtain those last two answers analytically?
The third value is -2*W(ln(2)/2)/ln(2), where W is Lambert’s W function (Google on that for some fun reading). I’ve seen an algebraic solution before, but it was very clever, and I don’t remember if it got the third value.
The “Lambert W function” is from Euler who named it after Lambert. It is specifically defined to be the solution to your equation. Other than that, there is no closed form solution (from this site ).
You will find that your problem is equivalent to solving the equation x[sup]x[/sup]=c.
If you find the answer unsatisfying, realise that, for instance, “sin” is also pretty much defined as the answer to an equation. It’s just that you’re used to those functions, so they feel like an answer, but the Lambert function doesn’t. Another answer is an explanation of why you can’t solve the equation in terms of familiar functions, but I left that in my other pants
Here is a thread that I started about something similar to this:
In it, you’ll find lots of explanations and references (and yes, Lambert’s Function gets mentioned too). I also believe that there have been other similar threads before the one I started.
Well, there’s a way to get around that, actually. The sine function can be defined in terms of the exponential function on the complex plane, which itself is “really” a map from C to T[sub]1[/sub]C to C again.
T[sub]1[/sub]C is the tangent (complex) line to C, and since C is a topological vector space this tangent space can be canonically identified with C itself. This is the first map.
The second map is the diffeomorphism from a ball around the origin in the tangent space of a complex manifold at the identity to a ball around the identity in the manifold itself. Since C has no conjugate points, the domain is the entire tangent space. This is the second map.
Now that defines exp: C —> C, and as we all know
sin(z) = (exp(iz)-exp(-iz))/2i
Here’s a quick little calculator function–f(x)= - sqrt(2)^x
Start with an x, and keep plugging the answer back in the function. It’ll converge to the third value ( -.76666469…) | crawl-data/CC-MAIN-2022-27/segments/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00647.warc.gz | null |
Introduction to our pages on Colonial History
American colonial history in New England and Bucklin family history are intertwined. As you read our pages on Colonial History, you will find about several noteable Bucklins in New England before 1800.
The first generations of Bucklins were in Massachusetts and Rhode Island. But the Bucklin family always included people who moved to new frontiers and were substantial contributors to the new societies they formed. (That’s why there are several places named Bucklin. When New York became available Bucklins went there. When the area that became Maine was cleared of the French, Bucklins went there. When the Erie Canal was opened to make it easy to go westward to new lands, an enterprising Bucklin had already gone into the wilderness, armed with a commission from the President of the United States to be a Justice of the Peace, and armed with his business skill there he sold the new arrivals land he had gathered at low prices.
During the period before 1800 a new and separate “American” character was formed
Not only was the character of individual residents of New England changed from that of being “European” to being “American,” formed, but also a new type of society and an entirely new type of government was formed.
“During …[the era of 1636 to 1790] major legal themes included the development of a body of internal law for the governance of a New World frontier commonwealth; the relationship between the colony and the mother country and the delineation of their respective powers; the establishment of intercolonial relations; the Americanization of the common law and its gradual replacement by local statute; the adjustment to the laws of trade and commerce under the mercantilist system, the formulation of the federal theory of empire and its corollary, dual sovereignty; the establishment of independence; the creation of a federal union under a national constitution…” [Conley 1998, at 9 – 10].
During New England’s colonial period the Bucklin family flourished and had significant local roles. When William Bucklin arrived in the Massachusetts Bay Colony, he was in the time period (1600 to 1799) of Massachusetts and Rhode Island on which we focus. Studying the Bucklin family during the successive generations of Joseph the 1st, Joseph the 2nd, Joseph the 3rd, Joseph the 4th and Joseph the 5th is a study of American Colonial History, and vice versa.
The American Revolution was in fact a value system revolution to the English and the Old World countries. The values and political philosophy owed much to the revolutionary thinking of Englishmen before the English Civil War. The foundations of American values were laid by the Englishmen who left for New England before the English Civil War. Here in the American colonies the social and political system was rooted in mavericks, innovation, risk-taking, impatience, vigorous intellectual arguments, a desire to move upward socially and economically, and great value was placed on actual constructive work by mind and body.
Probably the rich tradition of the Bucklin family of upholding one’s personal beliefs and the liberties of free persons was an important part of the reason why many Bucklins served in the Revolutionary Army. No doubt the fact that Joseph Bucklin stood a good chance of hanging for his shooting of the English navy ship captain (formally declared by a joint opinion of the English Attorney General and Solicitor General to be “treason”) was further impetus. When the Civil War came, Bucklins responded. A Medal of Honor of a Bucklin shines. But equally significant of the family tradition of upholding liberty are the Bucklin officer and the Bucklin first sergeant who volunteered to lead “Colored Troops” of the northern army when to do so was thought to be the way to dead end a military career. And so again, you can study Bucklins and learn American history; or you can study American History and run into Bucklins.
You can choose to go to:
- American History of the Massachusetts Bay Colony and the Rhode Island and Providence Plantations Colony, in the period 1600— 1799 (including history of the places Bucklins settled in that time frame and
- English History of England in the same time period (with particular reference to the Dorset area), and
- Gaspee Affair (with particular reference to the legal and political background) and the period of the early American Revolution (with particular reference to Massachusetts and Rhode Island).
This website has over 500 pages of information. We have much more information, but as a practical matter we have to limit the number of web pages we have to maintain. Thus our pages on colonial history are limited to pages which are of primary interest to persons interested in both Bucklin family and also American history. Thus we have, for example, pages on:
The development of the William Bucklin property in Pawtucket.
Although colonists came from many countries to the New World colonies, a combination of events propelled the culture and traditions of England to the forefront in the colonies. For example, with the settlement and early ship harbor of New York, the Dutch had a chance to stamp their culture on the new world. However, the peaceful surrender by the Dutch of New York to an English fleet of ships recognized that England, not Holland, ruled the seas between New York and Holland. The Dutch of New York adapted their commerce by adopting the English language and social customs of England. The French, after losing Quebec, receded to parts outside of what would become the original 13 states of the United States. The Spanish likewise, were outside the commercial area that became the original United States. And the Hanoverian line of the English kings that ruled England when the United States were born did not encourage German settlers to set up German areas in the colonies of the New World, rather encouraged Germans to be Americans under the rule of the German Hanoverian King George of England. | <urn:uuid:9a396683-8c9b-404d-a4f2-898fe2f1af48> | {
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Would you like to merge this question into it? MERGE already exists as an alternate of this question. Would you like to make it the primary and merge this question into it? MERGE exists and is an alternate of.
Yes, the CPU is the heart, or brain of any computer. The Central Processing Unit. It is the computer process chip that runs the compter Where is the CPU in a computer? The CPU is located under the heatsink that nearly allways has a fan on top of it! What does a CPU do in a computer? It is the brain of your computer and tells the computer what to do.
Why does a computer need a CPU? The CPU Central Processing Unit is needed because it is the mainpart of the computer where all the calculating and control activityis done. Without it, the computer will notoperate at all.
Why is the CPU called the brain of the computer? The CPU or Central Processing Unit is considered the brain of thecomputer because it is responsible for making decisions andcalculations.
The CPU chip contains millions of tiny transistorsthat work together. The CPU is the central processing unit of the computer. As it name - Central Processor Unit - implies, it is the circuitthat actually does the computing at binary level.
What does the CPU do for a computer? The control unit and arthematic unit of a computer are jointly known as the central processing unit CPU.
The CPU is the brain of any computer system. In human body all major decisions are taken by the brain and the other parts of the body function as diercted by the brain. Similarly, in a computer system, all major calculations and comparisons are made inside the CPU and the CPU is also responsible for activating and cntrolling the operations of other units of a computer.
How important is CPU important to the computer?
Without a CPU, a computer can do no work. Thus, without a CPU, there is no "computer. What is the functions of the CPU of a computer?
CPU stands for central processing unit. Overall a personal computer can be divided into the following major parts from point of view of functions performed: Input devices keyboard, mouse 2.
Output devices monitor which displays the output or results of work being done on the computer 3. CPU central processing unit, which is the brain of the computer which does all the processing and calculations on data inputs and provides output on screen or printer 4.
Storage devices memories, hard disk which store data. This is the box to which we connect monitor, keyboard, mouse and printer through cables. In technical terms the term CPU is used only for the processor, which fits on the motherboard.
Can you change the CPU speed on your computer? It is the main data processing component of your computer system. It is the "brain" of the computer.
What part of a computer is the CPU?
If you are using a desktop you will be able to locate it on the motherboard. Carefully remove your heat sink and you will find your cpu. What does a CPU meter on a computer do? It also tells you how much of your CPU is being used, for instance, you will run a program like halo or something that takes up a lot of room in your computer.Dissecting the Heart of Your Computer The central processing unit (CPU) is the heart of your computer.
This vital component, often referred to simply as the microprocesso r (or even just processor), is in some way responsible for every single thing your computer does.
Processor. Process-or that may likewise be understood as Central Processing Unit (CPU) can be really a processor, delegated with all the instructions of performing plausible I/O surgeries along with arithmetical alternatives of laptop or computer.
|what is processor in computer? Types of Microprocessor||Early CPUs were single core.|
|Features and Performance||In fact, one of my fellow editors highlighted a paragraph, sent it over to me in Slack, and basically said, "I have no idea what this means.|
A microprocessor or processor is the heart of the computer and it performs all the computational tasks, calculations and data processing etc.
inside the computer. Microprocessor is the brain of the computer.
In the computers, the most popular type of the processor is the Intel Pentium chip and the Pentium 1V is the latest chip by Intel . Heart of the computer is also known as the motherboard, that keeps the system moving. Like a heart that pumps the blood to keep the body alive, motherboard sends signals or allows electricity to go by the thin board to reach its destination or "components".
Jul 25, · Processor (computing) central processing unit (cpu), the hardware within a computer that executes (cpu) is electronic circuitry carries out instructions of program by performing basic this little. Dec 01, · Many scientific journels would say Processor is the heart of the computer, which is convincing.
But many of the text books wrongly say CPU is the heart of the computer. I would rather say, CPU is the whole body of the arteensevilla.com: Resolved. | <urn:uuid:31f01b5b-cd1a-4ffe-9077-55aa426da705> | {
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Sphere’s Equation
What is the equation for a sphere with a diameter of 6, and a center at (-1, 0, 1)?
Hint
The standard form of a sphere equation is:
$$(x-h)^{2}+(y-k)^{2}+(z-m)^{2}=r^{2}$$$Hint 2 In the standard form of a sphere equation, its center is at $$(h, k, m)$$ and $$r$$ is the radius. The standard form of a sphere equation is: $$(x-h)^{2}+(y-k)^{2}+(z-m)^{2}=r^{2}$$$
where the sphere’s center is at $$(h, k, m)$$ and $$r$$ is the radius.
Thus,
$$(x-(-1))^{2}+(y-0)^{2}+(z-1)^{2}=(\frac{6}{2})^{2}$$$$$(x+1)^{2}+(y-0)^{2}+(z-1)^{2}=(3)^{2}=9$$$
$$(x+1)^{2}+(y-0)^{2}+(z-1)^{2}=9$$\$ | crawl-data/CC-MAIN-2022-33/segments/1659882573667.83/warc/CC-MAIN-20220819100644-20220819130644-00484.warc.gz | null |
# Complex Numbers in Linear Algebra
• cowmoo32
In summary, complex numbers are defined as ordered pairs of real numbers and are denoted by C. They can be written as (a,b) where a is identified with the real number a and b is identified with i. The complex number (0,1) is denoted as i and has the property that i2 = -1. Addition and multiplication of complex numbers follow the operations of addition and multiplication of real numbers, except for the product which is defined as (a,b)*(c,d) = (ac-bd, ad+bc). This definition of multiplication leads to (0,1)*(0,1) = (-1,0) which shows that (0,1) is indeed i. Defining i as (
cowmoo32
I'm working my way through Shaum's Outline on linear algebra and in it they define a complex number as an ordered pair of real numbers (a,b). So given a real number a, its complex counterpart would be (a,0). Operations of addition and multiplication of real numbers work under the correspondence:
(a,0) + (b,0) = (a + b,0)
(a,0)*(b,0) = (ab,0)
I can follow that, but I'm confused how they define i. I know i=(-1)1/2. They define it as:
i2 = ii = (0,1)(0,1) = (-1,0) = -1
So am I to assume that any complex number written as (0,b) = -b?
The complex number (a,b) would be written a+bi. I think you might be getting confused because they defined i as i2 = -1, so, as you wrote, i = -11/2
The next step after defining i is writing a complex number z = a + bi. I'm just a little confused as to their definition of i.
Here is the first definition of complex numbers, verbatim:
The set of complex numbers is denoted by C. Formally, a complex number is an ordered pair (a,b) of real numbers.
We identify the real number a with the complex number (a,0); that is,
a<-->(a,0)
Thus, we view R as a subset of C, and replace (a,0) by a whenever convenient and possible
The complex number (0,1) is denoted by i. It has the important property that...
and then they go on to define i in the manner I posted above. After that,
Accordingly, any complex number z = (a,b) can be written in the form
z = (a,b) = (a,0) + (0,b) = (a,0) + (b,0)*(0,1) = a + bi
Ok, so as I typed that I see that (0,1) is simply i.
The one thing you did NOT say was how to multiply two such pairs- you only say "Operations of addition and multiplication of real numbers work under the correspondence:
(a,0) + (b,0) = (a + b,0)
(a,0)*(b,0) = (ab,0) "
The definition of the sum and product of two general such pairs is
(a, b)+ (c, d)= (a+ c, b+ d) ("component wise" addition)
(a, b)* (c, d)= (ac- bd, ad+ bd) (which is definitely not "component wise")
Your text should also show that all the usual "rules of arithmetic" ( addition and multiplication are associative and commutative and multiplication distributes over addition. There are additive and multiplicative identies, every pair has an additive inverse, and every pair except (0, 0) has a multplicative inverse).
From that definition of multplication (0, 1)*(0, 1)= (0*0- 1*1, 0*1+ 1*0)= (-1, 0). Since we are interpreting the pair (-1,0) to be the number "-1", that says that (0,1)*(0,1)= (0, 1)2= -1 and so (0, 1) is i. Given that we can then say that (a, b)= (a, 0)+ (0, b)= a(1, 0)+ b(0, 1)= a+ bi.
By the way, here we are defining i to be (0, 1), and then showing that i2= -1, not showing that (0, 1)= (-1)1/2. There are technical problems with "defining" i by "i=(-1)1/2". Every complex number, like every real number, has two square roots. Which of the two roots of -1 is "i"? The real numbers form an "ordered field" so we can distinguish between positive and negative square roots of numbers but the field of complex numbers is NOT an ordered field so it is impossible, a priori, to distinguish between the two roots of -1. Defining i to be the pair (0, 1) avoids that problem. (The other root of -1 is, of course, (0, -1)= -(0, 1)= -i. But to be able to call it "-i" we must be able to first distinguish that root from "i" itself.)
I would like to clarify and expand upon the concept of complex numbers in linear algebra. Complex numbers are a fundamental part of linear algebra and are represented by the set of numbers of the form (a, b), where a and b are real numbers and i is the imaginary unit defined as i = √(-1). Complex numbers are often used to represent quantities that cannot be expressed using only real numbers, such as solutions to quadratic equations.
In linear algebra, complex numbers are used to represent vectors and matrices. The real part of a complex number (a, b) is denoted as Re(a, b) and the imaginary part is denoted as Im(a, b). For example, the complex number (3, 4) has a real part of 3 and an imaginary part of 4.
The operations of addition and multiplication for complex numbers follow the same rules as for real numbers, with the added consideration of the imaginary unit i. For example, (a, b) + (c, d) = (a + c, b + d) and (a, b)(c, d) = (ac - bd, ad + bc). It is important to note that complex numbers do not follow the commutative property, meaning that the order of operations matters. For example, (a, b)(c, d) ≠ (c, d)(a, b).
Now, let's address the confusion about the definition of i. The symbol i is used to represent the imaginary unit, which is defined as i = √(-1). This means that i2 = -1. However, in linear algebra, complex numbers are represented as (a, b) and the imaginary unit is represented as (0, 1). Therefore, (0, 1)(0, 1) = (0*0 - 1*1, 0*1 + 1*0) = (-1, 0) = -1. This is why i is defined as (0, 1) in linear algebra.
In summary, complex numbers are a fundamental part of linear algebra and are represented by the set of numbers of the form (a, b). The imaginary unit i is defined as i = √(-1) and is represented as (0, 1) in linear algebra. The operations of addition and multiplication for complex numbers follow the same rules as for real numbers, with the added consideration
## What are complex numbers?
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (defined as the square root of -1). The real part of the complex number is a, and the imaginary part is bi.
## How are complex numbers used in linear algebra?
In linear algebra, complex numbers are used to represent vectors and matrices with complex elements. They are also used in solving systems of linear equations and in representing transformations in complex vector spaces.
## What is the difference between real and complex matrices?
Real matrices have only real numbers as elements, while complex matrices can have complex numbers as elements. In addition, the operations of addition, subtraction, and multiplication for complex matrices are defined differently than for real matrices.
## Can complex numbers be visualized geometrically?
Yes, complex numbers can be visualized geometrically using the complex plane, where the real axis represents the real part of the complex number and the imaginary axis represents the imaginary part. The magnitude of a complex number can also be represented as the distance from the origin to the point on the complex plane.
## What is the importance of complex numbers in quantum mechanics?
Complex numbers play a crucial role in quantum mechanics as they are used to represent physical quantities such as wave functions and probability amplitudes. They also play a role in understanding the behavior of particles at the quantum level and in making predictions about their behavior.
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1. ## Arc Length help!
I have a word problem im not sure how to solve regarding arc length
A radius of 93,000,000 miles measured between the earth and the sun at a central angle measuring 31 feet. Find the diameter of the sun(arc length in this case)
How do I solve with the central angle measured in feet?
2. Originally Posted by rusty1363
I have a word problem im not sure how to solve regarding arc length
A radius of 93,000,000 miles measured between the earth and the sun at a central angle measuring 31 feet. Find the diameter of the sun(arc length in this case)
How do I solve with the central angle measured in feet?
central angle from where? is there a diagram given?
it might help if you post the problem exactly as it is written.
3. yes there is a picture but ill write the text verbatim: At a time when the earth was 93,000,000 miles from the sun, you observe through a tinted glass that the diameter of the sun occupied an arc of 31'. Determine to the nearest ten thousand miles the diameter of the sun. Hint: Because the radius of arc AB is large and the central angle is small, the length of the diamater of the sun is approx. the length of the arc AB.
So the central angle is measured from the earth from the two radii that form the arc AB (the diameter of the sun).
4. Originally Posted by rusty1363
yes there is a picture but ill write the text verbatim: At a time when the earth was 93,000,000 miles from the sun, you observe through a tinted glass that the diameter of the sun occupied an arc of 31'. Determine to the nearest ten thousand miles the diameter of the sun. Hint: Because the radius of arc AB is large and the central angle is small, the length of the diamater of the sun is approx. the length of the arc AB.
So the central angle is measured from the earth from the two radii that form the arc AB (the diameter of the sun).
arc length formula ...
$s = r \cdot \theta$
$r$ = 93 million miles
$\theta$ = central angle in radians
fyi, ... 31' = 31 minutes of arc, not feet
you'll have to convert 31' to radians for $\theta$
5. ok thanks! | crawl-data/CC-MAIN-2017-22/segments/1495463607963.70/warc/CC-MAIN-20170525025250-20170525045250-00243.warc.gz | null |
MIDDLE SCHOOL EXPERIMENT: Colored Spectacles – An Alien Point of View
Light can be thought of as electromagnetic waves that travel through space with specific frequenciesand wavelengths. The light in the visible spectrum ranges from red to violet, like the colors of a rainbow. White light can be separated into the colors of the visible spectrum in passing through an object such as a prism or a water droplet. When the light passes through the object, each color is spread into a different angle so we are capable of seeing the colors. Wavelengths such as microwaves, radio waves, infrared, x-rays, and gamma rays are not visible to human beings. Remote sensors assist us in detecting these invisible electromagnetic wavelengths.
National Science Education Standards Addressed:
- Abilities Necessary to DO SCIENTIFIC INQUIRY/UNDERSTANDING About SCIENTIFIC INQUIRY
- PROPERTIES and Changes of Properties in MATTER/STRUCTURE and FUNCTION in living Systems
- ABIliTIES of Technological Design/UNDERSTANDING About Science and Technology
- PERSONAL HEALTH/Science and Technology in SOCIETY
- Six 35 mm x 130 mm pieces of heavy card stock with two 20 x 20 mm windows
- hole punch
- transparency squares (two each of clear, red, green, blue, yellow, and orange cut in 25 x 25 mm squares)
- two pieces of yarn 200 mm long
- one covered tray with objects of varying colors
- one sheet of white copy paper
- light socket
- light bulb
Take six pieces of the heavy card stock. Fold each end over 25 mm. Secure the flap with tape. Punch a hole in the middle of each folded, taped end. Tie one end of a strip of yarn to each end of the glasses. Make six pair of glasses, one each with red, green, blue, and clear lenses. Take one sheet of white copy paper and one covered tray to the table. Tape the white paper to the table. Place one experimental object at a time on the sheet. Assign one group member to be the recorder for your group. Tape the clear lenses to your glasses, observe each object, and describe what you see. Replace the clear lenses with red, green, and blue ones. Observe each object through each color. Record what you see.
- What are the similarities and differences of viewing each object through each type of lens? Develop asystematic way of describing and recording your results. Post your experimental results in the classroom.
- What might have happened if no white sheet was used? Select three other colors and determine how theresults may have been affected. Develop a systematic way of describing and recording your results.
- What happens to the experimental results if orange or yellow lenses are used? Would the experimentalresults be affected enough to justify the cost of adding orange or yellow lenses to the experiment? Why or why not?
- Take your glasses outside and look around. What lenses make clouds most easily visible compared withthe surrounding blue sky? What about trees or rocks?
- Why do scientists refer to red, blue, and green as primary colors? Why do artists refer to red, blue, andyellow as primary colors?
- Use the internet to investigate remote sensing, particularly passive remote sensing. How do theexperimental results relate to passive remote sensing with multispectral imaging?
- Use your experimental results to enhance the story line of your science fiction short story.
- Did anyone in your group see the colors of the objects differently from the majority of the group? Whichcolor(s) were involved? Why do some people see colors differently? Is this primarily a male or female trait?
- Overlap the red, blue, and green spectacle lenses in layers so that there is some overlap of all threecolors. There should also be some overlap in two-color combinations for each possible color pair. Some of the initial color should remain where no overlapping occurs. Shine the light from the bulb through them. What colors do you see in each area? Why do you see these colors? Prepare a chart detailing your results.
- Use the internet or books to learn about infrared and ultraviolet light and how such invisible light isuseful in remote sensing. What kind of light does your TV remote control use? Look at the light beam by pointing the remote control directly at a home video camera and pressing a button. Why does the light flash? | <urn:uuid:d2e71710-0470-4cfa-a01d-4061a3b463c5> | {
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This article is a comprehensive guide on Parts of Speech in English Grammar.
In English grammar, words are generally divided into eight different classes or Parts of Speech according to the work they do in a sentence. These eight classes are called Parts of Speech.
In short, there are eight different types of parts of speech is there, and those are:
As we already know that in English grammar there are eight basic parts of speech, let.s discuss them one by one.
A noun is specified as the name of a person, place or thing.
There are five kinds of nouns:
A proper noun is the name of a particular place or person. For example, Dubai is the richest city, here Dubai refers to the name of a place so it is a proper noun.
A common noun is specified as the name given in common to every person or thing. For example, The girl in my class.
A material noun denotes the matter of the substance of the thing. For example, the house is built of wood.
It is the name of a quality, action or state belonging to an object. For example, Darkness, movement, music, philosophy.
A collective noun is the name of a group of the collection of persons or things are taken together. For example, army, group, team, class, crowd.
For more details take a look at this article Noun in English Grammar.
A pronoun is a word that replaces, relates or which is used instead of a noun or equivalent.
Pronouns are classified into ten types:
It indicates any person while acting as a subject or an object. For example, I, we, they, you, he, she, him, her, our.
It indicates mainly non-living things. For example, it.
It demonstrates any particular sense. For example, this, that, these, those, it, so, such.
It distributes the sense of the subject or object. For example, each, every, either, neither.
It signifies the sense of the subject or object. For example, any, all, many, some, few, someone, anyone, none, anybody, nobody, everybody.
It reciprocates between two or among more than two subjects and makes a complementary sense. For example, each other, one another.
It makes an extra emphasis on the main subject and is constructed with s ‘self’ word. For example, my self, herself, himself, themselves, yourself.
It relates the subject or object with another clause or part of the sentence. For example, who, which, what, that, whose, whom, anyone, none, anybody.
It makes the sense of interrogation. For example, who, which, what, whom, whose.
It signifies a possession over any other person. For example, mine, ours, yours, his, its, theirs.
For more details check out my another article on Pronoun in English Grammar.
A verb is a word that states action, position or being.
There are seven types of verb:
These types of verb are restricted to the number and also to the persons. For example, I am a good boy.
Principle verbs are the main verb of a sentence, it carries the sense, action, or state of a sentence. For example, I played football yesterday. In this sentence "Play" sate an action that I performed yesterday.
These types of verb are often used alone, with one or more objects in a sentence. For example, Ram played cricket.
These types of verb do not allow with a direct object, that means you can not use this type of verb where an object is clearly mentioned. Example of this type of verb is 'River flows'
These types of verbs are used to form tense, mood, aspect, modality, voice, etc. For example, Ram taking a shower.
By its name we can say these type of verb is not finite, that means these types of verb is not show their tense.
In English grammar non-finite verb are three types:
If you want to learn about the verb in detail check out my another post Verb in English Grammar.
These types of adverbs are basically denoting time i.e. tomorrow, yesterday, etc.
This type of adverb denotes the way of doing things, i.e. slowly, fastly, lately, etc.
This type of adverb defines where the action of the verb happens, i.e. far, everywhere, etc.
These types of adverb define how often a thing happens, i.e. rarely, frequently, etc.
This type of adverb denotes a sentence that is true or in a negative sense or judgment, i.e. undoubtedly, certainly, etc.
This type of adverb is used when the sense of the statement is an interrogation, i.e. where, why, how, etc.
These types of adverb join sentences and clauses and also tell about the noun, i.e. why, which, etc.
These types of adverb are denoting the sense or intensity of a thing that is happening, i.e. fully, almost, etc.
These types of adverb modify the sentences, i.e. surely, luckily, etc.
If you want to learn about the adverb in detail check out my another post on Adverb in English Grammar.
An adjective is a word which qualifies a pronoun or a noun. There are eight kinds of adjectives.
To learn Adjective in detail check out my another post on Adjective in English Grammar.
A preposition is a word placed before a noun or pronoun to show its relation with other parts of speech in a sentence.
Prepositions are six types, those are:
Although prepositions are categorized in different ways like:
Check out preposition in details here: Preposition in English Grammar.
A conjunction is a word which is used to join words, phrases, clause, and sentences. There are three types of conjunction.
These types of conjunction join two sentences or clauses of the same kinds, i.e. but, like, etc.
These types of conjunction used with subordinate clauses, i.e. because, lest, if, etc.
These types of conjunctions used in pairs, i.e. neither-nor, either-or, so-as, etc.
More on conjunction click here: Conjunction in English Grammar.
An interjection expresses some sudden feeling of one’s mind. For example, Alas! We have lost the match. Hurrah! We won the match.
Some common interjections are Bravo, Hurrah, Alas, Oh, etc.
More on interjection: Interjection in English Grammar
So now I want to hear from you about your experience about this article, I hope you understand the topic Parts of Speech in English Grammar.
Please mention your doubts in the comment section, I will love to answer those questions. Also, I told you to give you the PDF downloadable link, you can find the download button below. | <urn:uuid:56cefa4c-e913-4d58-a687-08db00f26fb2> | {
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U.S. Department of Agriculture (USDA) scientists have completed the most comprehensive genetic analysis to date of the domesticated grape, applying new technology to uncover a surprising degree of genetic diversity and fine-tune genetic markers that may lead to grapes better equipped to resist pests and pathogens that now prompt repeated spraying of grape crops.
The study, published in the Proceedings of the National Academy of Sciences, shows that although wine and table grapes (Vitis vinifera) were domesticated up to 8,000 years ago in the Near East, they still have enough genetic diversity to offer untapped potential for developing desirable traits, according to lead author Sean Myles, now a postdoctoral research scientist at the Stanford University School of Medicine.
Other authors include geneticist Edward S. Buckler with the USDA-Agricultural Research Service (ARS) Robert W. Holley Center for Agriculture and Health in Ithaca, N.Y.; grape geneticist Christopher L. Owens at the ARS Grape Genetics Research Unit in Geneva, N.Y.; geneticist Mallikarjuna K. Aradhya at the USDA-ARS National Clonal Germplasm Repository in Davis, Calif.; horticulturist Bernard Prins, also at the Davis repository; Doreen Ware, a computational biologist affiliated with both the ARS Holley Center and Cold Spring Harbor; and collaborators from Cornell University, Stanford University, Cold Spring Harbor Laboratory and the University of Milan in Italy. Myles conducted the research as a postdoctoral scientist in Buckler's laboratory in Ithaca.
"Grapes are one of the world's most economically important fruit crops, and this study shows not only the potential for developing new approaches for improving existing varieties, but also the genetic relationships between many common varieties," said Edward B. Knipling, ARS administrator. ARS is the chief intramural scientific research agency of USDA.
The researchers say their results show that when breeders developed a successful wine or table grape variety, they were likely to continue planting it or its close relatives for centuries. As a result, grapes have experienced less intense breeding than other crops over the last millennia. That relative absence of crossbreeding has made grapes a natural target for many pests and pathogens, the researchers say. Many grape growers spend thousands of dollars each year spraying fungicides just to control powdery and downy mildews.
Grapes are woody perennial vines that take three years to mature from seedling to a fruit-bearing plant, so traditional breeding of new grape varieties is expensive and time-consuming. With the recent development of genomic tools for plant breeding, scientists worldwide have been searching for genetic markers associated with desirable traits in grapes. Scientists can use these markers to accelerate the development of grape varieties that are better equipped to resist diseases and pathogens, tolerate cold and drought, and offer the right mix of taste, maturity and other desirable traits.
In their study, the researchers found that domesticated grapes are likely to be sufficiently diverse to address many of the challenges faced by growers.
The researchers used a DNA microarray—a technology commonly used in genomics—equipped with 9,000 genetic probes to examine patterns of variation among pieces of DNA known as single nucleotide polymorphisms (SNPs) in 950 grape samples, also known as "accessions." Similar microarrays also have been developed to analyze the genomes of horses, cattle, sheep, corn and rice.
With the information generated from the custom microarray, the scientists developed a chart that outlines the genetic kinship of dozens of the grape accessions that produce some of the world's most popular wines, including Riesling and Pinot Noir.
The results also addressed confusion created by centuries of grape breeding and vegetative propagation. Over the years, when a grape variety sprouted a unique mutant trait or characteristic, such as a different fruit color, that new mutant was often vegetatively propagated and given a new name by a breeder. There are currently more than 10,000 names for grape cultivars worldwide but, from a genetic standpoint, some of the mutants that were named as new varieties were identical to their parents, or so nearly identical that genetic tests cannot distinguish between the mutant offspring and the parent, the researchers say.
As a result, Myles and his colleagues found that 58 percent of the 950 vinifera accessions they examined were so closely related that they appeared to be clones of at least one other accession. | <urn:uuid:e409e1a8-1820-47d0-811c-61c730b77c0c> | {
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# How Many Cups Is 18 Tbsp? Fast & Accurate Calculations
Ahhh, the age-old conundrum of converting tablespoons to cups — we’ve all been there. Whether you’re baking a batch of cookies or whipping up a delicious meal for dinner, it can be tricky trying to figure out just how many cups 18 tablespoons translates into. Everyday recipes often require exact measurements, so having the correct measurement is essential for making sure your dishes turn out perfectly every time. But how do you know exactly how many cups is 18 tbsp? That’s what this blog post sets out to answer. In this article, we will look at why understanding conversion measurements matter in cooking and explain step by step just how you convert from teaspoons and tablespoons into measuring cups so that all your culinary creations come out perfect each and every time.
## What is a tbsp?
A tablespoon (abbreviated as “tbsp”) is a unit of measurement often used in cooking and baking. It is equal to 1/2 fluid ounce, which is slightly less than a full measuring cup. In the United States, a tablespoon is usually equivalent to 3 teaspoons. When cooking or baking, it is important to use the correct measurement of ingredients to ensure that your recipes turn out perfectly every time. Knowing how to convert tablespoons into cups is essential for ensuring accurate measurements while cooking.
## What is a cups?
A cup is a unit of measurement used in cooking and baking. It is equal to 8 fluid ounces or 1/2 pint, and it is used to measure both dry and liquid ingredients. One cup is equivalent to 16 tablespoons, which is slightly more than a standard measuring cup. Knowing how many cups are in a tablespoon can help make sure your recipes are accurate and consistent every time.
## How many cups is 18 tbsp?
The answer is that 18 tablespoons equates to 1 1/4 cups. To convert from tablespoons to cups, divide the number of tablespoons by 16 (the number of tablespoons in a cup). For example, if you wanted to know how many cups are in 48 tablespoons, your calculation would look like this: 48/16 = 3 cups. So, for 18 tablespoons, 18/16 = 1.125 cups, which is equal to 1 1/4 cups.
## Benefit of knowing how many cups is 18 tbsp
Knowing how to convert tablespoons into cups is essential for ensuring accurate measurements while cooking. Having the right measurement is key in making sure that your recipes turn out just the way you want them to.
By knowing how many cups is 18 tbsp, you can make sure that you use the exact measurement of ingredients that your recipes require and that there is no guesswork involved. With the right measurements, your dishes will turn out consistently delicious each and every time.
Additionally, having the correct measurements can make meal planning and grocery shopping easier and more efficient, as you will know exactly how much of each ingredient you need to buy in order to complete your recipes.
## Conversion steps for cups is 18 tbsp
To convert 18 tablespoons into cups, simply divide the number of tablespoons (18) by 16 (the number of tablespoons in a cup). In this case it would look like this: 18/16 = 1.125 cups. This is equal to 1 1/4 cups. Once you know how to do the conversion from tablespoons into cups, it becomes much easier to adjust your recipes when needed.
Having a conversion chart on hand is an easy way to quickly convert tablespoons into cups without having to do any math. You can also use online calculators for quick conversions if you don’t have a chart handy. If you are measuring large amounts of ingredients, investing in a kitchen scale is another great option and will help make your recipes easier and more accurate.
## Conversion table for cups is 18 tbsp
In order to make the conversion from tablespoons to cups easier, here is a helpful table outlining how many tablespoons are equal to 1 cup:
• 2 tablespoons = 0.125 cups
• 4 tablespoons = 0.25 cups
• 6 tablespoons = 0.375 cups
• 8 tablespoons = 0.5 cups
• 10 tablespoons = 0.625 cups
• 12 tablespoon s= 0.75 cup
• 14 tablespoons = 0.875 cups
• 16 tablespoons = 1 cup
• 18 tablespoons = 1.125 cups
By having a conversion table like this handy, it can be much easier to figure out how many tablespoons you need in a recipe when conversions are required.
## Common mistakes when converting cups is 18 tbsp
When converting tablespoons to cups, it is common for people to forget that 16 tablespoons is equal to 1 cup. It’s important to remember that when you are measuring out your ingredients in order to get accurate measurements. Another common mistake is mixing up the units of measurement; make sure you are using teaspoons or tablespoons and not milliliters or ounces, as those are different measurements. Finally, when measuring dry ingredients such as flour or sugar, be sure to use the spoon-and-level method to make sure all of your measurements are accurate and consistent.
## Tips for accurately convert cups and 18 tbsp
When converting tablespoons into cups, it is important to measure accurately and use the correct unit of measurement. For instance, be sure to use teaspoons or tablespoons instead of milliliters or ounces. Additionally, when measuring dry ingredients like flour or sugar, make sure to use the spoon-and-level method for accuracy. Finally, having a conversion chart on hand can help make conversions much easier and faster, so you can get back to cooking in no time. By following these tips, you can easily convert tablespoons into cups for your recipes.
## Some recipes use cups and 18 tbsp
1. Chocolate Chip Cookies: This classic cookie recipe calls for both cups and tablespoons. The ingredients include 2 1/4 cups of all-purpose flour, 1 teaspoon of baking soda, 1 teaspoon of salt, and 2 sticks or 1 cup of butter, softened. It also calls for 3/4 cup granulated sugar, 3/4 cup packed brown sugar, 1 tablespoon vanilla extract, and 2 large eggs.
2. Zucchini Bread: This delicious bread recipe calls for both cups and tablespoons. The ingredients include 3 1/4 cups of all-purpose flour, 1 teaspoon baking soda, 1 teaspoon salt, 1 teaspoon ground cinnamon, and 1/2 cup vegetable oil. It also calls for 3 eggs, 2 cups grated zucchini, 1 cup granulated sugar, and 2 tablespoons of lemon juice.
3. Chocolate Cake: This moist and decadent cake recipe requires both cups and tablespoons. The ingredients include 3 cups of all-purpose flour, 1 tablespoon baking powder, 2 teaspoons baking soda, and 1 teaspoon salt. It also calls for 3/4 cup unsweetened cocoa powder, 2/3 cup vegetable oil, 1 1/2 cups granulated sugar, and 3 eggs.
By understanding how to convert tablespoons into cups, you can make sure your recipes turn out perfect every time! Measurement conversions are an important part of any successful culinary endeavor, so be sure to use accurate measurements when creating delicious treats in the kitchen. With knowledge of proper measurement conversions, you can quickly and easily make sure your recipes turn out perfect every time.
## Conclusion: How many cups is 18 tbsp
To answer the question of how many cups is 18 tbsp, the answer is 1 1/4 cups. Knowing how to convert tablespoons into cups can help ensure that your recipes turn out perfect every time. By having a conversion table handy, or by using online calculators and kitchen scales, you can make sure all of your measurements are accurate and consistent when creating delicious dishes in the kitchen.
## FAQ: Cups is 18 tbsp
### How many cups is 18 tablespoons of butter?
18 tablespoons of butter is equal to 1 1/4 cups. To convert from tablespoons to cups, divide the number of tablespoons by 16 (the number of tablespoons in a cup). So, for 18 tablespoons, 18/16 = 1.125 cups, which is equal to 1 1/4 cups.
### Is 1 cup equal to 18 tablespoons?
Learn how to convert tablespoons to cups with ease. One cup is equal to 16 tablespoons, so if you have 18 tablespoons, simply divide it by 16. The result is 1 and 1/8 cups. Accurate measurements are key for perfect recipe outcomes, so make sure to keep this in mind.
### How many cups is 18 tablespoons of powder?
18 tablespoons of powder is equal to 1 1/4 cups. To convert from tablespoons to cups, divide the number of tablespoons by 16 (the number of tablespoons in a cup). So, for 18 tablespoons, 18/16 = 1.125 cups, which is equal to 1 1/4 cups. It’s important to use accurate measurements when baking and cooking so that your recipes turn out perfectly every time.
### Is 18 tablespoons half a cup?
Discover the solution: 18 tablespoons equals 1 and 1/8 cups. This conversion is useful for accurately measuring ingredients in recipes.
### Is 18 tablespoons a 1/4 cup?
Don’t be fooled: 18 tablespoons does not equal a 1/4 cup. Let’s set the record straight: a full cup is the equivalent of 16 tablespoons. If you happen to have 18 tablespoons on hand, fear not. Just divide it by 16 and you’ll find yourself with 1 and 1/8 cups. Remember, precise measurements are crucial for flawless recipe results. Keep this in mind for culinary perfection.
### How many cups is 18 tablespoons of sugar?
Convert tablespoons to cups with precision by following this simple equation: divide the number of tablespoons by 16, the number of tablespoons in a cup. For example, 18 tablespoons will equal 1.125 cups, which is equivalent to 1 1/4 cups. Accurate measuring is crucial in baking and cooking to guarantee delicious results.
### Does 18 tablespoons equal 5 cups?
Incorrect Conversion: 18 tablespoons is not equivalent to 5 cups. However, to convert tablespoons to cups, simply divide the number of tablespoons by 16 (the equivalent number of tablespoons in a cup). In the case of 18 tablespoons, this will yield 1.125 cups, which can also be expressed as 1 1/4 cups. It is crucial to employ precise measurements when cooking and baking in order to achieve impeccable outcomes.
### Is 18 tablespoons a 3/4 cup?
18 tablespoons is not the same as 3/4 cup. To determine the number of cups in 18 tablespoons, divide the amount of tablespoons by 16 (the number of tablespoons in a cup). In the case of 18 tablespoons, this calculation will result in 1.125 cups, which can also be written as 1 1/4 cups. It is crucial to measure accurately when cooking and baking for the best outcome.
### How many cups is 18 tablespoons of milk?
To convert from tablespoons to cups, divide the number of tablespoons by 16 (the number of tablespoons in a cup). So, for 18 tablespoons, 18/16 = 1.125 cups, which is equal to 1 1/4 cups. When baking and cooking with milk or other liquid ingredients, it’s important to use accurate measurements for perfect results every time.
### How many cups is 18 tablespoons of solution?
To convert from tablespoons to cups, divide the number of tablespoons by 16 (the number of tablespoons in a cup). So, for 18 tablespoons, 18/16 = 1.125 cups, which is equal to 1 1/4 cups. When creating solutions for cleaning or other purposes that require precise measurements, it’s important to use accurate measuring tools for successful outcomes. | crawl-data/CC-MAIN-2023-40/segments/1695233506528.19/warc/CC-MAIN-20230923162848-20230923192848-00023.warc.gz | null |
Hardy (Feb. 7, 1877 – Dec. 1, 1947) was an English mathematician known for his work in number theory and mathematical analysis. Although Hardy considered himself a pure mathematician, he nevertheless worked in applied mathematics when he formulated a law that describes how proportions of dominant and recessive genetic traits will propagate in a large population (1908). Hardy considered it unimportant but it has proved of major importance in blood group distribution. As it was also independently discovered by Weinberg, it is known as the Hardy-Weinberg principle.
The Hardy-Weinberg equation | <urn:uuid:c426e804-170f-4229-92b2-1d20a0b677e8> | {
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- noun a group within a species with distinct characteristics
- noun deformation caused by stress
- Deformation of a material resulting from external loading. The measurement for strain is the change in length per unit of length.
- A lengthening, contraction, torsion, or other mechanical deformation resulting from an external force. Also called mechanical strain.
- verb to pour liquid through a sieve in order to separate out solids
- noun a condition in which a muscle has been stretched or torn by a strong or sudden movement
- noun a group of microorganisms which are different from others of the same type
- noun nervous tension and stress
- verb to remove impurities or solid matter from a liquid by passing it through a mesh
- verb to damage a part of the body through using it too hard or too much
Origin & History of “strain”
English has two distinct words strain. The older, ‘line of ancestry’ (OE), denotes etymologically ‘something gained by accumulation’. It comes from the prehistoric base *streu- ‘pile up’, which was related to Latin struere ‘build’ (source of English destroy, structure, etc). In the Old English period the notion of ‘gaining something’ was extended metaphorically to ‘producing offspring’, which formed the jumping-off point for the word’s modern range of meanings. Strain ‘pull tight, wrench’ (13th c.) was borrowed from estreign-, the stem form of Old French estreindre ‘pull tight, tie’. this in turn was descended from Latin stringere ‘pull tight, tie tight’ (source also of English strait, strict, and stringent (17th c.) and of a host of derived forms such as constrain (14th c.), prestige, restrain (14th c.) and constrict, district, restrict, etc). Strain ‘tune’ (16th c.) is assumed to be the same word, perhaps deriving ultimately from the notion of ‘stretching’ the strings of a musical instrument. | <urn:uuid:bc8ad074-bddb-45f6-a38a-0cd8b244d78b> | {
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1134. Armstrong Number
Description
Given an integer n, return true if and only if it is an Armstrong number.
The k-digit number n is an Armstrong number if and only if the kth power of each digit sums to n.
Example 1:
Input: n = 153
Output: true
Explanation: 153 is a 3-digit number, and 153 = 13 + 53 + 33.
Example 2:
Input: n = 123
Output: false
Explanation: 123 is a 3-digit number, and 123 != 13 + 23 + 33 = 36.
Constraints:
• 1 <= n <= 108
Solutions
Solution 1: Simulation
We can first calculate the number of digits $k$, then calculate the sum $s$ of the $k$th power of each digit, and finally check whether $s$ equals $n$.
The time complexity is $O(\log n)$, and the space complexity is $O(\log n)$. Here, $n$ is the given number.
• class Solution {
public boolean isArmstrong(int n) {
int k = (n + "").length();
int s = 0;
for (int x = n; x > 0; x /= 10) {
s += Math.pow(x % 10, k);
}
return s == n;
}
}
• class Solution {
public:
bool isArmstrong(int n) {
int k = to_string(n).size();
int s = 0;
for (int x = n; x; x /= 10) {
s += pow(x % 10, k);
}
return s == n;
}
};
• class Solution:
def isArmstrong(self, n: int) -> bool:
k = len(str(n))
s, x = 0, n
while x:
s += (x % 10) ** k
x //= 10
return s == n
• func isArmstrong(n int) bool {
k := 0
for x := n; x > 0; x /= 10 {
k++
}
s := 0
for x := n; x > 0; x /= 10 {
s += int(math.Pow(float64(x%10), float64(k)))
}
return s == n
}
• function isArmstrong(n: number): boolean {
const k = String(n).length;
let s = 0;
for (let x = n; x; x = Math.floor(x / 10)) {
s += Math.pow(x % 10, k);
}
return s == n;
}
• /**
* @param {number} n
* @return {boolean}
*/
var isArmstrong = function (n) {
const k = String(n).length;
let s = 0;
for (let x = n; x; x = Math.floor(x / 10)) {
s += Math.pow(x % 10, k);
}
return s == n;
}; | crawl-data/CC-MAIN-2024-33/segments/1722640447331.19/warc/CC-MAIN-20240805114033-20240805144033-00737.warc.gz | null |
Corals, like other living organisms, suffer from an increase in ocean acidity caused by rising levels of
An international team of scientists had intended to studythe impact that the long-term effects of high acidity of the ocean can have on three species of coral in the Caribbean. Researchers transplanted samples of corals in areas along the coastline of the Yucatan Peninsula in Mexico, where water from underwater sources lowered the pH of the surrounding sea water. According to the researchers, the environment in this area is even more acidic than predicted in 2100 in the oceans of the Earth.
Representatives of the species participated in the experiment.Siderastrea siderea, Porites asteoides and Porites porites. The first coped best with the changed conditions, while the survival rates of the second and third types decreased by 20% and 77%, respectively.
When all the surviving corals were able to grow, the researchers found that their skeleton density decreased by 15–30% compared with their relatives who live in other parts of the sea.
Earlier, biologists from the Lausanne Polytechnic School discovered a species of coral, which not only does not suffer from global climate change, but is also able to produce offspring. | <urn:uuid:fa18ecf3-cd4f-43f3-b722-6cde37b98671> | {
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} |
log b (0) is not defined. Any specific choice of value for 0 0 \frac00 0 0 will allow some function to be extended continuously. Therefore, we say division by zero is undefined. wheN we add 0 to any no. Thus, a whole number multiplied by zero equals zero, and vice versa. Solve 11 + 3x – 7 = 6x + 5 – 3x; First, combine like terms; then solve: as for e.g. The real logarithmic function log b (x) is defined only for x>0. Zero to the power of zero, denoted by 0 0, is a mathematical expression with no agreed-upon value.The most common possibilities are 1 or leaving the expression undefined, with justifications existing for each, depending on context. (ii) The place value of zero (0) is always 0. In a two-digit number, the place value of the ten-place digit is 10 times of the digit. 2. As, in 105, 350, 42017, 90218 the place value of 0 in each number is 0. In your textbook, a quadratic function is full of x's and y's.This article focuses on the practical applications of quadratic functions. There is no value! A parabola can cross the x-axis once, twice, or never.These points of intersection are called x-intercepts or zeros. It may hold any place in a number, its value is always 0. Any number times zero results in zero, it can never equal 2. Now let’s look at 0÷0. The multiplication property of zero is a little like the addition property in that it does not matter in what order you do the operation to the whole number. ln(0) is undefined. Initially, zero functioned as a mere placeholder—a way to tell 1 from 10 from 100, to give an example using Arabic numerals. We can't find a number x, so the base b raised to the power of x is equal to zero: b x = 0 , x does not exist. Why the natural logarithm of zero is undefined? Also explore many more calculators covering probability, statistics and other topics. zero gives different value to any no. Calculator to find out the standard score, also known as the z-score, of a normal distribution, convert between z-score and probability, and find the probability between 2 z-scores. Why log(0) is not defined. So the base b logarithm of zero is not defined. If zero is divided by a whole number, the quotient will be zero. There is no possible solution. For instance, if we mandate 0 0 = 1, \frac00=1, 0 0 = 1, then the function f (x) = x x f(x) = \frac xx f (x) = x x becomes continuous at x = 0. x=0. Such as 2+0=2 we gets the same digits as shown above so there is no value of 0 in addition . The graph of a quadratic function is a parabola. Since ln(0) is the number we should raise e to get 0: e x = 0. There is no number x to satisfy this equation. In algebra and combinatorics, the generally agreed upon value is 0 0 = 1, whereas in mathematical analysis, the expression is sometimes left undefined. Zero is a numerical value which (in "real life" or in the context of a word problem) might imply that there is "nothing" of something or other, but zero itself is a real thing; it exists; it is "something". Limit of the natural logarithm of zero. x = 0. The division property of zero is interesting. "That's not a full zero," Seife says. In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function, is a member of the domain of such that () vanishes at ; that is, the function attains the value of 0 at , or equivalently, is the solution to the equation () =.
2020 value of zero | crawl-data/CC-MAIN-2021-39/segments/1631780060908.47/warc/CC-MAIN-20210928214438-20210929004438-00520.warc.gz | null |
Although solid, the rocks of the Earth's mantle deform very slowly. Professor Patrick Cordier's team at the Materials and Transformation Unit (Université Lille, France) has developed a model that makes it possible, over timescales of several million years, to link the deformation of these rocks with mantle convection, the fundamental driver of plate tectonics. Until now, no experimental method in the laboratory had achieved the real conditions of deformation of mantle rocks. By applying this model to magnesium oxide, a solid present in the Earth's mantle, the scientists were able to show how atomic-scale defects in this mineral could be transmitted on a larger scale and over long periods of time.
Published in the journal Nature, these results call into question certain experimental approaches at high pressures and temperatures. They show that only a thin layer in the lowermost mantle can be regarded as a viscous liquid; elsewhere, the mantle behaves like a plastic solid.
The behavior of the Earth's mantle is chaotic on geological timescales. However, it appears relatively static, or indeed motionless, on the scale of a human lifetime (speeds in the mantle are comparable to fingernail growth speed).
To better understand how the deformation affecting rocks and minerals deep inside the Earth impacts mantle convection, a novel numerical approach has been developed by Patrick Cordier's team at the Materials andTransformation Unit.
By integrating theoretical concepts from solid-state physics and material deformation mechanisms, the scientists have been able to describe the behavior of minerals over previously inaccessible timescales and under experimentally unattainable conditions.
The researchers simulated the deformation of magnesium oxide (MgO), a solid naturally present in the lower mantle, under pressure and temperature conditions identical to those in the mantle (around a million times atmospheric pressure and a temperature of several thousand degrees). The researchers were thus able to observe the presence of defects on the atomic scale, called dislocations. For Cordier and his colleagues, such dislocations are the main cause of plastic deformation of the mantle, which is the fundamental driver of the Earth's heat machine (plate tectonics, volcanoes, earthquakes, etc).
From the perspective of geophysics, these results shake up some of the established notions in the field. To model mantle convection (the mechanism that releases the Earth's internal heat), the mantle is usually regarded as behaving like a viscous fluid over long timescales. In this study, the scientists show that only a thin layer in the lowermost mantle actually behaves in this way. Elsewhere in the mantle, the concept of viscosity does not apply, and the rock behaves like a plastic solid.
This new scientific data opens up a new research field in geophysics, linking the dislocation of solids on the atomic scale with fluid flows at the mantlescale.
More information: Modelling the rheology of MgO under Earth's mantle pressure, temperature and strain-rates. Patrick Cordier, Jonathan Amodeo, Philippe Carrez, Nature, 12 January 2012 | <urn:uuid:759befdb-5bcf-416d-b4e6-c2e356d26a5b> | {
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SummaryStudents learn about the physical force of linear momentum — movement in a straight line — by investigating collisions. They learn an equation that engineers use to describe momentum. Students also investigate the psychological phenomenon of momentum; they see how the "big mo" of the bandwagon effect contributes to the development of fads and manias, and how modern technology and mass media accelerate and intensify the effect.
Whether it is a truck, a washing machine or a compact disk writer mechanism, engineers commonly design products that move, so momentum is an important part of their design considerations. Over the years, engineers have been successful in using their knowledge of the force-momentum relationship to make vehicles safer in collisions. For example, vehicle front ends are especially designed to crumple, as a protective measure to reduce the forces felt by the occupants. Car safety seats are also designed to protect small children from accident forces.
After this lesson, students should be able to:
- Understand that linear momentum depends on both mass and velocity.
- Understand the difference between elastic and inelastic collisions.
- Appreciate why modern cars are safer in collisions than older ones.
- Understand how linear momentum can be described by an equation.
- Explain Newton's third law of motion.
More Curriculum Like This
This lesson introduces the concepts of momentum, elastic and inelastic collisions. Many sports and games, such as baseball and ping-pong, illustrate the ideas of momentum and collisions. Students explore these concepts by bouncing assorted balls on different surfaces and calculating the momentum for...
In this activity, students examine how different balls react when colliding with different surfaces. They learn how to calculate momentum and understand the principle of conservation of momentum.
Students examine how different balls react when colliding with different surfaces, giving plenty of opportunity for them to see the difference between elastic and inelastic collisions, learn how to calculate momentum, and understand the principle of conservation of momentum.
Students examine collisions between two skateboards with different masses to learn about conservation of momentum in collisions.
Each TeachEngineering lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.
All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN),
a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics;
within type by subtype, then by grade, etc.
Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.
All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.
- Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
- Fluently divide multi-digit numbers using the standard algorithm. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
- Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
- Solve linear equations in one variable. (Grade 8) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
- Solve real-world and mathematical problems involving the four operations with rational numbers. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
- Use mathematical expressions to describe the movement of an object (Grade 8) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
- Recognize that mathematical models are used to predict orbital paths and events (Grade 8) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback!
Have you ever been at a grocery store and seen a shopping cart run loose through the parking lot? Hopefully it wasn't heading straight for your car — but if it were, would you rather a fully loaded cart or an empty cart hit your car? Probably the empty cart! An empty cart would not cause as much damage if it hit your car because it has less momentum. Momentum is the measurement of an object's mass multiplied by how fast the object is moving.
Momentum can move from one object to another object when they bump into each other. The movement of momentum from one object to another is called transfer of momentum. When a fully-loaded shopping cart collides with the side of a car, you can see evidence of momentum transfer— the car is dented!
In this lesson, we will explore the idea of momentum, studying momentum transfer by examining collisions. It is important for engineers to understand momentum transfer so they are able to design safe cars, investigate accidents, plan the way spaceships dock with space stations, and all sorts of other things. A good understanding of collisions and momentum is also an excellent way to improve your bowling score or a game of pool, too!
Lesson Background and Concepts for Teachers
Momentum, which is given the symbol p , is a combination of the mass and velocity of something that is moving. Mathematically, momentum is described by the equation:
p = m x v
where: m = mass of the object in kilograms
v = velocity of the object in meters per second
In this equation, the p and v are in bold because momentum and velocity are considered vector quantities. That means that they have both a magnitude and direction.
Understanding momentum can lead to some surprising answers to questions. For example, consider the question "If a BB bumped into a bowling ball, would the bowling ball move?" The answer to the question depends upon how much momentum the bb has. If the bb was not going very fast it would not have much momentum and the bowling ball would not move very much (you probably could not even measure any motion in most cases). If the BB was going very fast, though, it would be a different story. If a bb that weighed 57 grams (about 2 oz.) were moving at 355 meters per second (almost 800 miles per hour, a bit faster than the speed of sound), and it hit a bowling ball that weighed 4.5 kg (about 10 lbs.), the bowling ball would roll away at 4.5 meters per second (about 10 mph)! Through the collision, the momentum of the little bb moving very fast is transferred to the bowling ball, which moves slower because it has much more mass!
Elastic and Inelastic Collisions
Collisions cause momentum to move from one object to another object. In everyday life, collisions occur all over the place — pool games, traffic accidents, rubber balls bouncing, baseballs being hit by bats, and more. You can probably observe many collisions just by looking around a classroom. Understanding momentum gives engineers an insight to understand different kinds of collisions. This understanding can help make cars safer, predict the results of two objects bumping into each other, or examine the evidence of a traffic accident.
There are different kinds of collisions. Sometimes objects bump into each other then bounce away from each other, such as when a rubber ball hits the ground. Engineers call this kind of collision an elastic collision. Other times, objects that bump in to each other stick together, such as when a ball of play dough hits the ground – splat! Engineers call these kinds of collisions inelastic collisions. Most of the time, collisions are part elastic and part inelastic. For example, when a shopping cart hits a car, it might dent the car (an inelastic collision), but it also bounces off of the car (an elastic collision). We can learn more about momentum by examining different types of collisions.
An example of a "perfect" elastic collision would be if you dropped a rubber ball on a hard sidewalk and it bounced back to its original height. In real life, balls do not bounce back all the way up to their original height because they lose some of their energy when they hit the ground. This energy may be lost through the creation of a noise (boing!) or through a very small change in temperature (due to the release of energy when the ball collides with the floor). Another example of an elastic collision is when two balls bump into each other on a pool table. In this case, the balls do not stick together — they bounce off each other, even though some energy is lost when the balls make a noise.
An inelastic collision occurs when objects bump into each other and stick together. An example is when two train cars are getting hooked together. The engine of the train pushes one car until it bumps into another car and they hook together. Then, the two cars roll away, connected, at a slower speed.
In both elastic and inelastic collisions, the total momentum of all the objects before the collision is the same as the total momentum of all the objects after the collision. The fact that momentum is not lost is called the Law of Conservation of Momentum. The Law of Conservation of Momentum helps us predict what happens when things bump into each other. For example, during a pool game, if the 8-ball is hit directly with the cue ball, the cue ball will stop and the 8-ball will roll with as much momentum as the cue ball had before the collision. Since the masses of the two balls are the same, this means that the 8-ball will have the same velocity as the cue ball had. If the 8-ball is hit on its side, the two balls will roll in different directions, but with a total combined momentum equal to what the cue ball had before the collision. In other words, even though both balls may be moving, they will move at a slower speed than the cue ball was moving by itself because the cue ball has transferred some of its momentum to the 8-ball.
When a fast-moving car hits a telephone pole, there is a tremendous amount of force between the front bumper and the pole. The force can be calculated by the force-momentum relationship:
F = Δp/Δt
Where Δp = change in momentum
(Note: The Δ symbol is called "delta," and represents change)
Δt = the time it took for the change to occur
Why would engineers be interested in this relationship? One reason is to make cars in accidents be safer for people. This relationship says that if momentum is transferred over a longer period of time, there is less force involved. If the force of a collision can be reduced, the chances that someone would get hurt in an accident are lower. Therefore, if engineers can figure out a way to increase the time required for a car to come to a stop in a collision, they can lower the forces that will impact people riding in the car, and the people will be less likely to be hurt. In fact, during the many years of car design, engineers have been very successful in accomplishing this! Older cars were built more solidly than today's cars; their front ends would not crumple in an accident. When an older car ran into something solid, it stopped very quickly, and so both the driver and the car experienced a large impact. Engineers have designed newer cars to crumple on impact, lengthening Δt and thus reducing the force experienced by the occupants. You could say that newer cars are safer in accidents than older cars because of an understanding of the force-momentum relationship.
Newton's Third Law of Motion
In 1687, Isaac Newton first published his now-famous laws of motion. Newton's third law of motion can be summarized as: for every action, there is an equal and opposite reaction. How does this apply to collisions? When an object, such as a car, strikes a telephone pole (the action), the pole strikes back on the car, causing the car to stop and/or sustain damage (the reaction). So, if you are designing a safer car, you will probably aim to reduce the size of the car's force on telephone poles, other cars, etc. (the action) so that the car's occupants suffer less force from the reaction.
Conservation of momentum: A situation in which the total momentum of all the objects before a collision equals the total momentum of all the objects after a collision.
Elastic collision: A collision in which objects bounce off each other. No energy is lost in an ideal elastic collision.
Force-momentum relationship: The force in a collision is equal to the change in momentum divided by the change in time. A large force is required for an object to lose momentum quickly (such as a car stopping quickly when it hits a stationary object).
Inelastic collision: A type of collision in which objects stick together. Some energy is lost in an inelastic collision due to occurrences such as noise, breaking glass, bending metal.
Momentum: A combination of the mass and velocity of a moving object.
Newton's third law of motion: For every action, there is an equal and opposite reaction.
- Skateboard Disaster - Students explore the concepts of momentum and the conservation of momentum by examining collisions between skateboards.
- The Big Mo - Momentum is not only a physical principle; it is also psychological phenomenon. Students learn how the "big mo" of the bandwagon effect contributes to the development of fads and manias, and how modern technology and mass media accelerate and intensify the effect. Students develop media literacy and critical thinking skills to analyze trends and determine the extent to which their decision may be influenced by those who manipulate a few opinion leaders.
Ask the students to explain momentum and give examples of objects that exhibit linear momentum. Ask them to predict how a small object with a large speed bumping into a small object would affect the large object. (Answer: Because momentum would be transferred, the large object would move, but at a slower speed than the small object.) Ask them to predict what would happen if a large object such as a bowling ball bumped into a small object like a marble and transferred all its momentum. (Answer: The small object would move away from the large object at a high speed.) Ask the students what the force-momentum relationship is and how engineers would use it. (Answer: The force-momentum relationship describes the amount of force required for a change in momentum. It is equal to the change in momentum divided by the change in time. Engineers might use it to predict how much force is required to slow something down, speed something up, or determine how much force would be exerted upon people in a car crash.)
Discussion Questions: Solicit, integrate and summarize student responses.
- Could a bee fly fast enough crash into and stop a moving bowling ball? (Answer: It depends on the size and speed of the bowling ball.)
- What sports exhibit examples of momentum? (Possible answers: Pool/billiards, bowling, curling, shuffleboard.)
- Why are engineers interested in momentum? (Possible answer; to design safer vehicles, etc.)
Voting: Ask a true/false question and have students vote by holding thumbs up for true and thumbs down for false. Count the votes, and write the totals on the board. Give the right answer.
- True or False: A bee flying at the speed of sound could stop a 10-lb. bowling ball rolling slowly? (Answer: True. It could stop a bowling ball rolling up to 10 mph.)
- True or False: New cars crumple in accidents because engineers decided to use cheaper materials. (Answer: False. They were designed to crumple as a way to lengthen the time of the collision and thus, decrease the forces felt by the occupants.)
Lesson Summary Assessment
Have students practice calculating momentum and force:
- If the mass of an object is 15 kg, and it is moving at 7m/s, what is its momentum? (Answer: p=15 kg * 7 m/s= 105 kg m/s = 105 N s)
- What is the change in momentum of an object when a 12 N force is applied to the object for 10 seconds? (Answer: ∆p=F*∆t=12 N * 10 S=120 kg m/s=120 N s)
Have students solve the following problem:
A 1,500-kilogram car travelling at 10 meters per second (about 22 mph) strikes a parked car on the side of the road.
- If the car comes to a stop in half of a second, what force is exerted on the parked car by the moving car? What force is exerted on the moving car by the parked car? [Answer: F = Δp/Δt = (1,500 kg x 10 m/s) / 0.5 s = 30,000 N (the force is the same because of Newton's third law of motion – the action and reaction are equal!)]
- If the force on the cars is to be kept under 10,000 N, what minimum time is necessary for the car to come to a complete stop? [Answer: F = Δp/Δt Δt = Δp/F = (1, 500 kg x 10 m/s) / 10,000 N = 1.5 seconds]
Bingo: Provide each student with a sheet of paper containing a list of the lesson vocabulary terms. Have each student walk around the room and find a student who can define one vocabulary term. Students must find a different student for each word. When a student has all terms completed s/he shouts "Bingo!" Continue until two or three students have bingo. Ask the students who shouted "Bingo!" to give definitions of the vocabulary terms.
Lesson Extension Activities
Have students research the force-momentum relationship and find out how it is related to Newton's Second Law.
Scavenger Hunt: Have the students find two objects that collide either elastically or inelastically. If applicable, have the students bring these objects into the classroom.
Elastic and inelastic collisions:http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html.
Pytel and Kiusalaas. Engineering Mechanics Dynamics. Pacific Grove, CA: Brookes/Cole Publishing Company, 1999.
ContributorsChris Yakacki; Ben Heavner; Malinda Schaefer Zarske; Denise Carlson
Copyright© 2004 by Regents of the University of Colorado.
Supporting ProgramIntegrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder
The contents of this digital library curriculum were developed under a grant from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education and National Science Foundation GK-12 grant no. 0338326. However, these contents do not necessarily represent the policies of the Department of Education or National Science Foundation, and you should not assume endorsement by the federal government.
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What is Mammography?
Mammography is a type of imaging that uses a low-dose x-ray system to examine breasts. A mammography exam, called a mammogram, is used to aid in the diagnosis of breast diseases in women.The American College of Radiology and the Society of Breast Imaging both recommend mammography as a means of saving lives. A study in the journal Cancer also found that regular mammography screening can save lives.
An x-ray (radiograph) is a painless medical test that helps physicians diagnose and treat medical conditions. Radiography involves exposing a part of the body to a small dose of ionizing radiation to produce pictures of the inside of the body.
We offer screening mammography (and other breast studies and procedures) for men, as well as women.
Digital mammography is superior to film mammography because digital images can be processed by a computer and displayed in multiple formats, as well as being able to be easily manipulated. In some cases film images are then digitized, but at the S. Mark Taper Foundation Imaging Center the image is taken digitally, so there is no loss in resolution. This type of mammogram is called direct full-field digital mammography.
To schedule an appointment, please call (310) 423-8000.
What is Computer-Aided Detection (CAD)?
We are proud to be among the first to offer computer-aided detection (CAD) technology, a recent advance in the field of breast imaging. This state-of-the-art system is becoming the national standard of practice. CAD enhances the images obtained from a digitally acquired Mammogram, and can aid in the detection of any breast abnormalities, highlighting “regions of interest” such as abnormal areas of density, mass or calcification. In clinical studies, CAD has been shown to increase the detection of breast cancer.
What are Some Common Uses of a Screening Mammogram?
Mammograms are used as a screening tool to detect early breast cancer in women experiencing no symptoms. They are also used to detect and diagnose breast disease in women experiencing symptoms such as a lump, pain or nipple discharge.
Mammography plays a central part in the early detection of breast cancers because it can show changes in the breast up to two years before a patient or physician can feel them. Current guidelines from the U.S. Department of Health and Human Services (HHS), the American Cancer Society (ACS), the American Medical Association (AMA) and the American College of Radiology (ACR) recommend a screening mammography every year for women, beginning at age 40. Research has shown that annual mammograms lead to early detection of breast cancers, when they are most curable and breast-conservation therapies are available.
How is the Procedure Performed?
During mammography, a specially qualified radiologic technologist will position your breast in the mammography unit. Your breast will be placed on a special platform and compressed with a paddle. The technologist will gradually compress your breast.
Breast compression is necessary in order to:
- Even out the breast thickness so that all of the tissue can be visualized.
- Spread out the tissue so that small abnormalities won't be obscured by overlying breast tissue.
- Allow the use of a lower x-ray dose since a thinner amount of breast tissue is being imaged.
- Hold the breast still in order to eliminate blurring of the image caused by motion.
- Reduce x-ray scatter to increase sharpness of picture.
You will be asked to change positions slightly between images. The routine views are a top-to-bottom view and an oblique side view. The process will be repeated for the other breast.
You must hold very still and may be asked to keep from breathing for a few seconds while the x-ray picture is taken to reduce the possibility of a blurred image. The technologist will walk behind a wall or into the next room to activate the x-ray machine.
When the examination is complete, you will be asked to wait until the technologist determines that the images are of high enough quality for the radiologist to read.
The examination process should take about 30 minutes.
For more information or to make an appointment, please call (310) 423-8000.
The S. Mark Taper Foundation Imaging Center provides a full range of advanced imaging, both radiology and cardiology, as well as interventional radiology and interventional tumor (oncology) treatments to the greater Los Angeles area, including Beverly Hills, Encino, Mid-Cities, Sherman Oaks, Silver Lake, Studio City, Toluca Lake, and West Hollywood. | <urn:uuid:a2809e5a-c398-41a2-9c38-9f4967dd90e4> | {
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Image of the Sun from SOHO
Click on image for full size
Courtesy of NASA
SOHO Catches Glimpse of the Sun's "Far Side"
News story originally written on June 23, 1999
The Solar and Heliospheric Observatory (SOHO) caught a rare view of the far side of the Sun. Scientists can now see if a solar storm is coming before it reaches Earth. This may save the satellite industry millions of dollars each year.
When the Sun releases large amounts of energy, the light makes patches of hydrogen gas glow. This glow is invisible to Earth, but not to SOHO. This new technology can give scientists a few days warning before the storm actually hits.
SOHO also captured the largest shadow ever seen. When Comet Hale-Bopp passed by in 1997, SOHO took a few photographs. Behind the comet, was a shadow over 150 million kilometers long. When the comet came near the Sun, it developed a long tail made of hydrogen. This tail and the comet itself were projected onto the sky.
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Gaps between teeth can be a concern in terms of form and function. Often the best way to close spaces between teeth is with braces. Other methods exist, but it depends on your desired outcome. Read on to find out how we can help you achieve your ideal smile.
What is Diastema? (Gap Teeth)
Diastema is a space between two teeth, often the upper front teeth (midline), but gaps can occur between any two teeth.
Gaps between teeth occur for a variety of reasons. Gaps are common in children with baby teeth, but often the space between teeth disappear as the larger adult teeth come through. Not everyone has a full set of adult teeth, so the missing tooth may be a cause of the gap.
Oversized gums or enlarged tissue at the top of the gum above the front teeth (called a labial frenum) can also cause a gap between teeth. Minor surgery can reduce the problem tissue. Some people have a gap between their teeth because their jaw bone is larger than their teeth. Actions such as thrust tongue or thumb sucking place pressure on the upper front teeth causing them to move apart.
Diastema is a type of malocclusion, the misalignment of teeth or bite. Other forms of malocclusion include overbite, underbite, crossbite, crowded teeth, crooked teeth and overlapping teeth.
Some adults notice gaps developing between their teeth as they mature. The concern may be a reflection of gum problems which reduces the support for the teeth, causing them to splay out. This may also be seen in teenagers with more severe forms of gum disease.
The Problem with Gaps Between Teeth
There are a few issues with gaps between teeth that may cause a person to consider closing the gaps with orthodontic treatment.
Gaps between teeth may be closed for reasons of form and function. The main problem with gaps is that gums are unprotected while chewing on food. Hard foods like potato chips and crusty bread can harm soft gums.
Plaque deposits can build up between teeth with gaps and lead to bad breath and tooth decay. Periodontal (gum) disease is more likely to occur in unprotected gums. Over time, periodontal disease can lead to loose teeth in the gums and in severe cases can cause teeth to fall out.
If spaces are developing as a result of gum disease, it will be essential for you to treat the gum disease before the orthodontist can close the spaces with a variety of appliance types such as braces or clear aligners. Your orthodontist can examine you and determine if the gaps are developing secondary to gum disease.
Considering Orthodontic Treatment as an Adult?
Get the smile you’ve always wanted with our innovative adult orthodontic treatment! Improve your smile and book an appointment!
In many people, a gap between teeth isn’t a problem and there is no medical reason for closing the gap. Some people don’t like the look of the gaps between their teeth. The gaps cause them to feel self-conscious when they talk or smile, and they prefer the look of teeth touching each other.
Orthodontic Treatment Options for Gaps
There are various orthodontic treatments for closing the gaps between teeth including traditional and lingual (inside) braces, Invisalign plus other dental options.
Braces with Elastic Chains
Braces work by pulling teeth into place with brackets, wire and elastic chain. Small elastic bands linked in a chain can move teeth together and close gaps. The coloured elastic chain fits over the brackets attached to teeth.
Elastic chains can close gaps in as little as six weeks to six months. Power chains are often the last step in braces as other problems such as teeth alignment are treated before closing the gaps.
The time patients require braces and elastic chains can vary based on several variants, including:
Care of Elastic Chains
To ensure your elastic chains are as effective as possible and treatment time is as short as possible, follow these care guidelines:
Brushing and flossing as directed by your Orthodontist
Don’t eat hard, sticky foods
Try to stop biting your nails or pen
Aesthetic Braces Options
If you’re worried about the look of metal braces, there’s the option of clear or lingual braces. Clear braces work the same as metal braces with the metal brackets replaced with translucent ceramic brackets that blend with teeth. Clear braces are more expensive than metal braces.
Lingual braces are worn on the back of your teeth so they can’t be seen. While the aesthetics of lingual braces are excellent, there are drawbacks to consider.
Each of the brackets for lingual braces must be custom-fit to the tooth, so they’re a more expensive option than metal and clear braces. You may need to wear lingual braces slightly longer than traditional braces; they don’t have the same power to pull teeth into place. Lingual braces can cause a lisp which reduces with practice speaking.
Invisalign is an option for some patients with gaps, but for others they’re not an option. Much of it depends on the size of the gap. Small gaps can be closed with the clear plastic aligners however if the gap is large, aligners may not pull the teeth together as much as required and the patient needs braces.
Some people may choose a combination of Invisalign aligners and bonding or veneers to close a large gap. Veneers and bonding are long-term options, but they aren’t lifelong like braces.
Closing gaps between teeth is one of the more simple orthodontic cases compared to other types of malocclusion.
If you would like to know your options, contact us online to make an appointment. | <urn:uuid:6c26f98c-2faf-465e-8397-4a0ef52f7484> | {
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Place Value Cards and Holder
I am joining up with a great group of mathematics bloggers for a new series of blog hops–Fly on the Math Teacher’s Wall. ย Love that title, don’t you? ย Each blog hop will focus on a central math topic, and you’ll find blog posts across all grade levels discussing the same topic. ย You’ll also find some *freebies* along the way! ย The topic this first time around is place value.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally recommend.
I’ve been working with my struggling 2nd graders on place value this week, and here are the three big ideas I would like to share:
1. Take it slowly
2. Use multiple representations
Let’s take a look at each idea individually.
Take it slowly
Bottom line, you’ve got to make sure students understand tens and ones before going on to hundreds or thousands. ย If students don’t have a strong understanding of tens and ones, they won’t get hundreds or thousands. ย If you are working with students, even in the upper grades, who are struggling with place value, you’ve got to go back to tens and one. ย Here is a great quote from Kathy Richardson’s book, How Children Learn Number Concepts:
“However, if place value concepts are to be meaningful, children need to know more than what digit is in the tens place and what digit is in the ones place. ย They need to know that the underlying structure of two-digit numbers is based on organizing numbers into groups of tens and ones. ย This understanding is critical and basic to successful future work with larger numbers and decimals.”
It is a major milestone for a child to transition from counting items one-by-one to understanding that we can count a group of ten as a single unit. ย This skill is call unitizing. ย Students should have ample experience working with numbers from 10 to 20 and seeing that numbers can be thought of as ten and some moreย before going on to greater 2-digit numbers. ย If a student can’t tell you immediately that a ten and 3 ones is 13, they probably need more practice with numbers less than twenty.
Use multiple representations
There are so many great materials for modeling place value concepts, and students benefit from being exposed to a variety of representations. ย Many primary teachers use bundles of straws to show tens and ones as they count the number of days in school, and that’s certainly a great representation. ย But Richardson shares a story in her book that shows how a rote routine can mask the fact that there is little to no understanding of the underlying concept. Think of turning that calendar process around. Picture this: the students enter the room on the 76th day of school only to find that their carefully bundled straws are now in a pile on the floor! ย Their task is to help you put the display back together by counting out groups of ten. :).
Another great way to count the days in school is to use ten-frames. ย I found this picture on Pinterest a while back, and I love it!
Van de Walleย tells us that traditional base-10 blocks are far too abstract for students at the beginning of their place value journey. Why? ย Because students can’t physically combine ones to make the ten and they can’t break it back into ones. ย Instead, students should use groupable materials, such as linking cubes, bundled straws, beans in small cups, etc., so they can build a ten and then break it back into ones.
Another great tool for exploring place value is the hundred chart. Because the structure of the chart groups numbers into rows of tens, it is a useful representation for exploring concepts such as ten more and ten less.As students are representing 2- and 3-digit numbers with concrete materials, be sure to connect their hands-on learning to the abstract symbols behind the models they are building. ย This handy little holder and set of place value cards helps students see the values behind (literally…ha ha) the digits. I made it to show numbers through the hundreds, but of course you would start with only tens and ones.
Let’s go back to how you might use this with the straw mess from the 76th day of school. ย As students count out bundles of tens, use the cards with the multiples of tens to keep count of how many straws they have. ย So when they count 10 straws and bundle them, put out the card that says 10. ย Don’t put it in the holder yet–just use the cards to keep track. ย As they bundle another ten, replace the 10 card with the 20 card. ย Keep doing this until they are showing the 70 card to represent the 70 straws that are now bundled. ย They count the final 6 straws and take out the 6 card. ย Now put the 70 card and the 6 card in the holder so they can see how the 70 bundled straws are shown as 7 tens.
Of course there are lots of other uses for the cards. ย You can grab your own set of cards and holder for free by clicking here!
Over the years, I think we’ve fallen into a routine when it comes to the questions we ask students to answer about place value. ย There are the old reliable questions like What digit is in the tens place?ย or What is the value of the 7 in this number?ย There’s nothing wrong with those questions, but they don’t really assess understanding. ย A correct answer to the first question only indicates that a student has memorized the place value names and positions; it does not tell you whether or not they understand that a 3 in the tens place means 3 groups of ten or that 3 tens is the same as 2 tens and 10 ones. ย One way to make your questioning more powerful is to simply add more to your question. ย For example, after asking a student to tell you the digit in the tens place, follow-up with one of these extensions:
• what does that mean?
• can you show me that with … (linking cubes, base-10 blocks, etc.)?
• can you draw that for me?
• how is that different from having the same digit in the ones place?
So just remember to dig a little deeper to make sure there is understanding below the surface.
Head on over to the next stop along the hop and see what The Recovering Traditionalist has to say about place value!
Similar Posts
1. Julia says:
Thanks so much for sharing! I always LOVE your ideas and use many of your materials. I am a 5th/6th grade math interventionist near San Antonio (SCUC ISD) and we are currently working on place value of bigger numbers. Would you be willing to share your schedule/how you organize your time with your kiddos? I only have about 40 minutes with each group so I’m definitely pinched for time. I’d be so grateful!!! [email protected]
1. Donna Boucher says:
Julia, I see each of my groups for 30 minutes a day, five days a week. It’s a pull-out program during what we call “extended learning” time, so the kiddos aren’t missing classroom instruction. My groups are from 3-5 students.
2. The Math Spot says:
I love your discussion of asking the “right” questions. I especially like “How is that different from having the same digit in the ones place?” This could be an especially powerful question for 4th and 5th grade students who are expected to understand patterns of ten on the place value chart and each place value being 10x more or 10x less than the place value to the left or right. Thanks for the thought provoking post! The Math Spot
1. Donna Boucher says:
Thanks so much! Questioning can definitely transform a good lesson into a great one.
3. TheElementary MathManiac says:
I love your point about base 10 blocks being to abstract for kids when they first start learning place value! The importance of multiple representations can not be emphasized enough!
Tara
The Math Maniac
1. Donna Boucher says:
Exactly, Tara! If we only use one type of manipulative, we run the risk of students beginning to think that “ten” has to look a certain way.
4. Sarah M says:
Love the points you make in this post! So important to use various tools depending on student readiness. Thanks so much for sharing!
Sarah
1. Donna Boucher says:
My pleasure, Sarah! I think multiple representations lead to deeper understanding.
5. Dianne Leoni says:
Hi Donna,
Just worked with a first grader and love the additional resources. I am heading to the copier right now to run off the place value holder and cards. I think to see the multiples of 10 and then 1s will help.
Your comments are spot on regarding assessment and manipulatives. It really bothers me that some publishers simply ask which digit is in the tens place, which digit is in the ones place? As if the response measures place value understanding. Add in the student has a 50-50 chance of getting it correct just with a guess.
Wish I could get PD or grad credits for reading your posts. I would have my PHD by now!!! | crawl-data/CC-MAIN-2024-26/segments/1718198861696.51/warc/CC-MAIN-20240617024959-20240617054959-00617.warc.gz | null |
Intelligence: The Measurement of Cognitive Capabilities
Most people think of intelligence as describing "how smart" someone is. However, the actual definition is quite a bit more complicated than that. Psychological researchers and theorists have actively debated and argued over how to best define and measure intelligence for over one hundred years. Individual theorists and researchers have disagreed on which mixture of cognitive skills and mental capacities (problem solving, abstract thinking, creativity, memory, concentration, interpersonal skills, body/movement skills, etc.) should be included within the definition, and how to measure these important attributes in a fair, culture free manner. At present, intelligence is best thought of not as a single ability or attribute, but rather as a global construct encompassing many different and separate cognitive abilities. According to the American Psychological Association, intelligence describes a person's ability to understand complex ideas, to adapt to the environment, to learn from experience, and to engage in reasoning and decision-making in all sorts of situations (both new and familiar).
History of Intelligence Testing and IQ Score
One way to understand the complexity involved in defining intelligence is to look at how tests measuring this construct have evolved over time. The first scientific test of intelligence, constructed by Alfred Binet during the early 1900's, was designed to provide French educators with a reliable method for discriminating special needs children from the general school population for purposes of classroom placement. Binet used children's test scores across a series of tests to separate children who needed special education classes from youth who could function well in regular classes. Binet's approach attempted to measure children's "general mental ability" by assessing different facets of their reasoning and thinking abilities and then using these scores to predict the learning environment likely best suited to each child.
In 1905, Binet and colleague Theodore Simon updated Binet's previous test to create the Binet-Simon scale, again for the purpose of identifying students in need of special education. The primary advance with this second intelligence test was that the Binet-Simon score was computed so as to take into account each student's chronological age. Since children's raw abilities and capacities typically advance and expand as children develop over time, it had become apparent that it was impossible to talk about intelligence without taking age into account. Without taking age into account, a truly smart child would appear (but not actually be) less intelligent than a less cognitively gifted adult simply because the adult is more cognitively mature and experienced than the child. Consequently, in computing the intelligence of children, it is vital that ability comparisons be made in comparison to similarly aged people, and not to all people of all ages.
Stanford University psychologist Lewis Terman released the "Stanford Revision of the Binet-Simon Scale," (now known as the Stanford-Binet and still in use today) in 1916. This test defined intelligence in terms of four separate cognitive factors:
- verbal reasoning (e.g., the ability to solve verbal problems and to demonstrate language mastery through demonstrations of vocabulary knowledge and sentence comprehension)
- quantitative reasoning (e.g., the ability to solve math problems)
- abstract/visual reasoning (e.g., the ability to solve problems requiring comprehension of complex relationships between geometric shapes)
- short-term memory (i.e., the ability to hold facts in memory for a short period of time).
Terman's test also resulted in a comprehensive score that he called an "intelligence quotient"; what we call today an "IQ". In the manner of the Binet-Simon scale, each student's IQ score was computed based on a mathematical comparison of his or her tested mental age with performance expectations based on chronological age.
Today, the most commonly administered IQ test for children in the middle childhood stage of development is the Wechsler Intelligence Scale for Children, forth edition (WISC-IV) test, originally developed by David Wechsler in 1974, and last revised in 2003. The WISC-IV measures general intelligence (providing an overall IQ score) and also two broad cognitive factor scores: a verbal IQ score (measured with sub-tests that require listening and answering examiner questions, and measuring comprehension, vocabulary, and general information items), and a performance IQ (measured by timed problems that require children to physically manipulate puzzles, pictures, blocks, etc., rather than providing verbal answers to questions).
Though intelligence is measured with an eye to multiple competencies and takes a broad rather than narrow view of these abilities, there do appear to be several specific underlying cognitive abilities that strongly influence individuals' global intelligence. Children's speed of cognitive processing (i.e. their reaction time, or how quickly they can solve mental tasks) and their ability to use cognitive strategies effectively (i.e., how quickly they can select and use effective problem-solving strategies) strongly influence their measured global intelligence, with quicker, more efficient problem-solving children tending to have higher IQ scores. Cognitive speed and efficiency are considered aspects of children's information processing abilities.
There are a few take-home messages we hope to convey here. One is that intelligence is defined not as a measurement of a single monumental ability, but rather as multidimensional construct taking into account measures of a broad array of abilities and talents. Another is that many of the sub-tests comprising modern IQ tests are heavily culture-bound and draw upon children's past experiences and knowledge they have previously learned. It is quite difficult, if not impossible, to measure intelligence in a pure raw form, separate from those things that intelligence enables people to accomplish, such as learning vocabulary or manipulating puzzles comprised of familiar shapes. | <urn:uuid:f93da6e6-f202-41c0-b7b2-35886abcbd53> | {
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Type of Unit: Concept
Prior Knowledge
Students should be able to:
Write and evaluate simple expressions that record calculations with numbers.
Use parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols.
Interpret numerical expressions without evaluating them.
Lesson Flow
Students learn to write and evaluate numerical expressions involving the four basic arithmetic operations and whole-number exponents. In specific contexts, they create and interpret numerical expressions and evaluate them. Then students move on to algebraic expressions, in which letters stand for numbers. In specific contexts, students simplify algebraic expressions and evaluate them for given values of the variables. Students learn about and use the vocabulary of algebraic expressions. Then they identify equivalent expressions and apply properties of operations, such as the distributive property, to generate equivalent expressions. Finally, students use geometric models to explore greatest common factors and least common multiples.
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Students use a rectangular area model to understand the distributive property. They watch a video to find how to express the area of a rectangle in two different ways. Then they find the area of rectangular garden plots in two ways.Key ConceptsThe distributive property can be used to rewrite an expression as an equivalent expression that is easier to work with. The distributive property states that multiplication distributes over addition.Applying multiplication to quantities that have been combined by addition: a(b + c)Applying multiplication to each quantity individually, and then adding the products together: ab + acThe distributive property can be represented with a geometric model. The area of this rectangle can be found in two ways: a(b + c) or ab + ac. The equality of these two expressions, a(b + c) = ab + ac, is the distributive property.Goals and Learning ObjectivesUse a geometric model to understand the distributive property.Write equivalent expressions using the distributive property.
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Algebraic Reasoning
Type of Unit: Concept
Prior Knowledge
Students should be able to:
Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.
Lesson Flow
This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.
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Algebra
Mathematics
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Pearson
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Students use the distributive property to simplify expressions. Simplifying expressions may include multiplying by a negative number. Students analyze and identify errors that are sometimes made when simplifying expressions.Key ConceptsThis lesson focuses on simplifying expressions and requires an understanding of the rules for multiplying negative numbers. For example, students simplify expressions such as 8 − 3(2 − 4x). These kinds of expressions are often difficult for students because there are several errors that they can make based on misconceptions:Students may simplify 8 − 3(2 − 4x) to 5(2 − 4x) because they mistakenly detach the 3 from the multiplication.Students may simplify 8 − 3(2 − 4x) to 8 − 3(−2x) in an attempt to simplify the expression in parentheses even though no simplification is possible.Students may simplify 8 − 3(2 − 4x) to 8 − 6 −12x. This error could be based on a misunderstanding of how the distributive property works or on lack of knowledge of the rules for multiplying integers.Goals and Learning ObjectivesSimplify more complicated expressions that involve multiplication by negative numbers.Identify errors that can be made when simplifying expressions.
Subject:
Algebra
Material Type:
Lesson Plan
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# m07, #10
Author Message
VP
Joined: 18 May 2008
Posts: 1260
### Show Tags
04 Nov 2008, 22:13
3
KUDOS
2
This post was
BOOKMARKED
A welder received an order to make a 1 million liter cube-shaped tank. If he has only 4x2 meter sheets of metal that can be cut, how many metal sheets will be required for this order? (1 cubic meter = 1,000 liters)
A. 92
B. 90
C. 82
D. 78
E. 75
Source: GMAT Club Tests - hardest GMAT questions
SOLUTION IS HERE: m07-72451-20.html#p1189373
SVP
Joined: 17 Jun 2008
Posts: 1547
### Show Tags
04 Nov 2008, 23:16
2
KUDOS
1 million liter cube shaped tank = 1000cubic meter cube shaped tank = cube shaped tank with side = 10 m.
Thus, surface area of the tank = 6*10*10 = 600 square m.
Area of one sheet = 4*2 = 8 sqaure m.
Hence, number of sheets required = 600/8 = 75.
VP
Joined: 18 May 2008
Posts: 1260
### Show Tags
05 Nov 2008, 03:19
I cld not understand how cum Surface area is 6*10*10?
scthakur wrote:
1 million liter cube shaped tank = 1000cubic meter cube shaped tank = cube shaped tank with side = 10 m.
Thus, surface area of the tank = 6*10*10 = 600 square m.
Area of one sheet = 4*2 = 8 sqaure m.
Hence, number of sheets required = 600/8 = 75.
CIO
Joined: 02 Oct 2007
Posts: 1218
### Show Tags
05 Nov 2008, 03:23
How many facets does a cube have? It has 6 facets. In this case each facet will have the area of 10*10 meters. This is why we need 6*10*10=600 square meters of metal.
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VP
Joined: 18 May 2008
Posts: 1260
### Show Tags
05 Nov 2008, 04:35
Oh my god! HOw can i do this g8 blunder? Im losing control on my sense of understanding as test date is coming closer
dzyubam wrote:
How many facets does a cube have? It has 6 facets. In this case each facet will have the area of 10*10 meters. This is why we need 6*10*10=600 square meters of metal.
Manager
Joined: 29 Jun 2009
Posts: 51
### Show Tags
18 Feb 2010, 10:59
ritula wrote:
A smith received an order to make a 1 million liter cube-shaped tank (note that 1 cubic meter = 1,000 liters of water). If he has only 4 meter by 2 meter sheets of metal that can be cut, how many metal sheets will be required for this order if the smith has to make the sides first and then weld them?
(A) 92
(B) 90
(C) 82
(D) 78
(E) 75
[Reveal] Spoiler: OA
E
Source: GMAT Club Tests - hardest GMAT questions
I found this one tricky, not because of the maths but because of the physics!!.
I assumed the volume as 1 million litres instead of 1000 cubic metres.
However, a quick google search revealed a fundamental problem in my understanding.
Volume is "usually" measured in cubic metres and the capacity in litres.. Although IMHO the value in litres also tells us the volume in a way...
Here are the links I read - 1) http://en.wikipedia.org/wiki/Volume- Although it says that "Volume is commonly presented in units such as cubic meters, cubic centimeters, liters, or milliliters.", it clarifies a little below that "Volume and capacity are sometimes distinguished"
2) http://en.wikipedia.org/wiki/Litre
Manager
Joined: 09 Dec 2009
Posts: 122
### Show Tags
18 Feb 2010, 11:06
ritula,
I do stuff like that all the time. It usually means I'm getting I:
A. Have been spending too much time studying and have begun to overthink/overanalyze and it actually starts to take away from my ability to solve questions. You get paranoid looking for tricks so much so that obvious things get lost in the shuffle. When this happens its time for a break. Atleast an hour or two, sometimes the rest of the night, sometimes a full day. Just get your mind off of it.
B: If I'm well rested it means I'm not focused enough. There is no subsitute for full concentration. Just break it down into bit sized pieces and let the answer come to you.
Hope that helps, even a bit.
_________________
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Round 2: 07/10/10 - This time it's personal.
Intern
Joined: 24 Oct 2009
Posts: 7
### Show Tags
18 Feb 2010, 14:01
Ah, like the surface area trick.
I took a different approach (and ended up using more time, so not recommended).
For a cube to fit 1000 cubic meter of water, each side should be 10 m in length. That means a cube with dimensions 10x10x10 is needed.
Or, each face with 10x10 dimensions. So the question is, how many plates would be required for making one face ? If you draw a basic figure, it would come out to be 12.5 . Since we have 6 faces, 12.5 * 6 = 75, Hence answer E.
Cheers,
naheed
Intern
Joined: 08 Oct 2009
Posts: 12
### Show Tags
19 Feb 2010, 02:51
Take only one side 10*10=100.
Now metal sheet 4*2=8
it is clear one side take 13 sheets
so 13*6=78
Am i right
CIO
Joined: 02 Oct 2007
Posts: 1218
### Show Tags
19 Feb 2010, 05:28
No. You've rounded up the answer. The key here is that we can cut the sheets however we need. You have to divide the whole area (600) by the sheet area (8) to get the right answer (75).
saqibbaig wrote:
Take only one side 10*10=100.
Now metal sheet 4*2=8
it is clear one side take 13 sheets
so 13*6=78
Am i right
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Intern
Joined: 31 Jan 2010
Posts: 6
### Show Tags
20 Feb 2010, 16:57
@saqibbaig
I too did the same mistake in the first go.
i.e calculated the number of sheets needed for each side
(10*10)/(4*2) = 12.5 so took it as 13 sheets per side hence 13*6 = 78 sheets
Now if you read the question again, it asks for how many sheets are needed, No restriction on cutting the sheets.
Or you can think like this. 12.5 sheets for 1 side.. so 25 sheets for 2 sides hence 75 sheets for 6 sides.
Or simply do 600/8.
Hope this helps.
Intern
Joined: 08 Oct 2009
Posts: 12
### Show Tags
21 Feb 2010, 07:12
I do agree but last sentence(if the smith has to make the sides first and then weld them) is making confusion.Question is clear prior to these words.
Intern
Joined: 18 Feb 2011
Posts: 10
### Show Tags
23 Feb 2011, 12:16
2
KUDOS
Need to build a 1,000,000 liter cube tank.
Information given (1 m^3 = 1,000 liters)
Step 1) We need 1,000,000 liters, so we use the formula above to get 1,000 m^3.
Now we forget about liters completely and focus on 1,000 m^3.
Step 2) To get volume = length x width x depth. The tank is cubed, which means that all sides should be equal. So cubed root of 1,000 m^3 = 10 m. Now we have our length = 10m, width = 10 m, and depth = 10m.
Step 3) To get the area of one side = length x width = 10m x 10m = 100m^2.
Imagine a six sided die, all sides are 100m^2.
Now to find out how many sheets are required, the problem changes to a surface area problem. Since the sheets will go on the outside (surface area) of the cube.
Step 4) 6 sides x 100m^2 = 600m^2.
Step 5) Sheets are 4m x 2m = 8m^2.
Step 6) 600m2/8m^2 = 75 sheets
Manager
Joined: 03 Nov 2009
Posts: 65
### Show Tags
23 Feb 2011, 13:37
This got me thinking took me some time to do it by drawing ..
By doing that we have 12.5 Plates per Face = 12.5 * 6 = 75
scthakur wrote:
1 million liter cube shaped tank = 1000cubic meter cube shaped tank = cube shaped tank with side = 10 m.
Thus, surface area of the tank = 6*10*10 = 600 square m.
Area of one sheet = 4*2 = 8 sqaure m.
Hence, number of sheets required = 600/8 = 75.
simple and short Good One.
Intern
Joined: 30 Nov 2011
Posts: 30
Location: United States
GMAT 1: 700 Q47 V38
GPA: 3.54
### Show Tags
28 Feb 2012, 09:07
Hi, for those who got D an an answer, which I also got, the mistake is that we have to multiply 6 times 12.5 instead of 6 times 13. The reason is that the surface area of each side of the cube is 100 and needs 12 sheets (each of an area of 8) so the remaining 4 m2 (100 - 96) could be filled up with half a sheet's area, so 12.5 x 6 faces = 75
Math Expert
Joined: 02 Sep 2009
Posts: 39661
### Show Tags
27 Feb 2013, 06:19
1
KUDOS
Expert's post
ritula wrote:
A smith received an order to make a 1 million liter cube-shaped tank (note that 1 cubic meter = 1,000 liters of water). If he has only 4 meter by 2 meter sheets of metal that can be cut, how many metal sheets will be required for this order if the smith has to make the sides first and then weld them?
(A) 92
(B) 90
(C) 82
(D) 78
(E) 75
[Reveal] Spoiler: OA
E
Source: GMAT Club Tests - hardest GMAT questions
BELOW IS REVISED VERSION OF THIS QUESTION:
A welder received an order to make a 1 million liter cube-shaped tank. If he has only 4x2 meter sheets of metal that can be cut, how many metal sheets will be required for this order? (1 cubic meter = 1,000 liters)
A. 92
B. 90
C. 82
D. 78
E. 75
1 million liter cube-shaped tank = 1,000,000/1,000 = 1,000 cubic meter tank, which means that we have the cube-shaped tank with a side of 10m (10^3=1,000).
Surface area of this tank would be 6 (# of faces of a cube) * 10^2 (area of each face) = 600 m^2.
# of 4*2=8m^2 sheet needed is 600/8=75.
_________________
Intern
Joined: 15 Jan 2012
Posts: 1
### Show Tags
27 Feb 2013, 16:25
1 million cubic liter water to be stored. So, capacity of cube should be 1000 cum. Side of cube = 10m. Surface area =6*10*10=600sqm. Number of sheets =Surface area of cube/surface area of 1 sheet = 600/8=75. Its E.
Intern
Joined: 10 Aug 2012
Posts: 19
Location: India
Concentration: General Management, Technology
GPA: 3.96
### Show Tags
28 Feb 2013, 01:09
3
KUDOS
$$Since, 1000 litres = 1 m^3$$
$$1000000 litres = 1000 m^3.$$
$$Tank is cube and volume is 1000 m^3, so each side will be 10 meter$$
$$Surface Area of Cube = 6* 10m * 10m = 600m^2.$$
$$Area of single sheet = 4m * 2m = 8m^2$$
$$Total number of sides = 600m^2/8m^2 = 75 sheets$$
Intern
Joined: 27 Feb 2013
Posts: 5
### Show Tags
28 Feb 2013, 08:14
something different from Bunuel...we may to have different language for same kind of questions in PS...so friends need to convert language in numeric contents effectively and efficiently...concentration is the key to success....
Senior Manager
Joined: 07 Sep 2010
Posts: 327
### Show Tags
18 May 2013, 23:35
Bunuel wrote:
A welder received an order to make a 1 million liter cube-shaped tank. If he has only 4x2 meter sheets of metal that can be cut, how many metal sheets will be required for this order? (1 cubic meter = 1,000 liters)
A. 92
B. 90
C. 82
D. 78
E. 75
Hi Experts,
Could you please explain the above RED colored part. What does it mean - "1 million liter cube-shaped "?
_________________
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Re: m07, #10 [#permalink] 18 May 2013, 23:35
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# m07, #10
Moderator: Bunuel
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | crawl-data/CC-MAIN-2017-26/segments/1498128320338.89/warc/CC-MAIN-20170624203022-20170624223022-00046.warc.gz | null |
There Lies Treblinka (Treblinke Dort)
Lyrics by: Anonymous
Composer: Eduardo Bianco (tango “Oración”)
Performed by Frieda Bursztyn Radasky
Portrait of Frieda Bursztyn Radasky taken in Turkheim, Germany, 1946.
Jews assembled under guard before deportation from Warsaw. Warsaw, Poland, July-September 1942.
Zydowski Instytut Historyczny Instytut Naukowo-Badawczy
Frieda Bursztyn Radasky learned There Lies Treblinka in 1943 while working in the kitchen at a coal depot in the Praga district of Warsaw, outside the ghetto area. The kitchen workers, mostly young women, witnessed countless Jews being deported from the ghetto. Many deportees believed the Nazi propaganda that the trains were headed to work camps, where survival was possible. Radasky and her coworkers knew the trains led to death camps. There Lies Treblinka was their way of acknowledging the horrible truth.
According to Radasky, There Lies Treblinka was written over a period of time with each worker contributing to the lyrics. The song survives in a number of variant forms; Radasky recorded her version around 1990 during an oral history session with her daughter, whose voice can occasionally be heard on the recording.
USHMM recorded sound archive | <urn:uuid:83dfdc5b-e1bd-4992-ab5b-2a73af7a6a6b> | {
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"int_score": 4,
"language": "en",
"language_score": 0.9196183085441589,
"score": 3.546875,
"token_count": 295,
"url": "https://www.ushmm.org/collections/the-museums-collections/collections-highlights/music-of-the-holocaust-highlights-from-the-collection/music-of-the-holocaust/there-lies-treblinka"
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Posted on Categories:abstract algebra, 抽象代数, 数学代写
# 数学代写|抽象代数代写Abstract Algebra代考|Math417 Coset Decoding
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## 数学代写|抽象代数代写Abstract Algebra代考|Coset Decoding
There is another convenient decoding method that utilizes the fact that an $(n, k)$ linear code $C$ over a finite field $F$ is a subgroup of the additive group of $V=F^n$ . This method was devised by David Slepian in 1956 and is called coset decoding (or standard decoding). To use this method, we proceed by constructing a table, called a standard array. The first row of the table is the set $C$ of code words, beginning in column 1 with the identity $0 \cdots 0$. To form additional rows of the table, choose an element $v$ of $V$ not listed in the table thus far. Among all the elements of the coset $v+C$, choose one of minimum weight, say, v’. Complete the next row of the table by placing under the column headed by the code word $c$ the vector $v^{\prime}+c$. Continue this process until all the vectors in $V$ have been listed in the table. [Note that an $(n, k)$ linear code over a field with $q$ elements will have $|V: C|=q^{n-k}$ rows.] The words in the first column are called the coset leaders. The decoding procedure is simply to decode any received word $w$ as the code word at the head of the column containing $w$.
I EXAMPLE 11 Consider the $(6,3)$ binary linear code
$$C={000000,100110,010101,001011,110011,101101,011110,111000} .$$
The first row of a standard array is just the elements of $C$. Obviously, 100000 is not in $C$ and has minimum weight among the elements of $100000+C$, so it can be used to lead the second row. Table 29.4 is the completed table.
## 数学代写|抽象代数代写Abstract Algebra代考|Historical Note
In this “Age of Information,” no one need be reminded of the importance not only of speed but also of accuracy in the storage, retrieval, and transmission of data. Machines do make errors, and their non-man-made mistakes can turn otherwise flawless programming into worthless, even dangerous, trash. Just as architects design buildings that will remain standing even through an earthquake, their computer counterparts have come up with sophisticated techniques capable of counteracting digital disasters.
The idea for the current error-correcting techniques for everything from computer hard disk drives to CD players was first introduced in 1960 by Irving Reed and Gustave Solomon, then staff members at MIT’s Lincoln Laboratory….
“When you talk about CD players and digital audio tape and now digital television, and various other digital imaging systems that are coming-all of those need Reed-Solomon [codes] as an integral part of the system,” says Robert McEliece, a coding theorist in the electrical engineering department at Caltech.
Why? Because digital information, virtually by definition, consists of strings of “bits”-0’s and 1’s – and a physical device, no matter how capably manufactured, may occasionally confuse the two. Voyager II, for example, was transmitting data at incredibly low power-barely a whisper — over tens of millions of miles. Disk drives pack data so densely that a read/write head can (almost) be excused if it can’t tell where one bit stops and the next 1 (or 0 ) begins. Careful engineering can reduce the error rate to what may sound like a negligible level – the industry standard for hard disk drives is 1 in 10 billionbut given the volume of information processing done these days, that “negligible” level is an invitation to daily disaster. Error-correcting codes are a kind of safety net-mathematical insurance against the vagaries of an imperfect material world.
In 1960, the theory of error-correcting codes was only about a decade old. The basic theory of reliable digital communication had been set forth by Claude Shannon in the late 1940s. At the same time, Richard Hamming introduced an elegant approach to single-error correction and double-error detection. Through the 1950s, a number of researchers began experimenting with a variety of errorcorrecting codes. But with their SIAM journal paper, McEliece says, Reed and Solomon “hit the jackpot.”
## 数学代写|抽象代数代写Abstract Algebra代考|Coset Decoding
$$C=000000,100110,010101,001011,110011,101101,011110,111000 .$$
## 数学代写|抽象代数代写Abstract Algebra代考|Historical Note
1960 年，欧文·里德 (Irving Reed) 和古斯塔夫·所罗门 (Gustave Solomon) 首次提出了从计算机硬盘驱动器到 CD 播放器的当前纠错技术的想法，然后是麻省理工学院林肯实验室的工作人员……。
“当你谈论 CD 播放器和数字音频磁带以及现在的数字电视和即将到来的各种其他数字成像系统时，所有这些都需要 Reed-Solomon [代码] 作为系统的组成部分，”Robert McEliece 说，他是一名加州理工学院电气工程系的编码理论家。
1960 年，纠错码理论只有大约十年的历史。克劳德·香农 (Claude Shannon) 在 20 世纪 40 年代后期提出了可靠数字通信的基本理论。与此同时，Richard Hamming 引入了一种优雅的单错误纠正和双错误检测方法。整个 1950 年代，许多研究人员开始试验各种纠错码。但 McEliece 说，凭借他们的 SIAM 期刊论文，里德和所罗门“中了大奖”。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 | crawl-data/CC-MAIN-2024-30/segments/1720763517833.34/warc/CC-MAIN-20240722064532-20240722094532-00853.warc.gz | null |
# Can someone tell me why this happens?
I observed this awkward repetition while making a strategy for the game "odd-eve". Can someone tell me why this happens?
Let $${ a }_{ 1 }$$ be the sum of digits of $$x$$ in base 10.
Let $${ a }_{ n}$$ be the sum of digits of $${ a }_{ n-1}$$ in base 10, for integers $$n>1$$.
Let $$g\left( x \right) =\lim _{ n\rightarrow \infty }{ { a }_{ n } }$$
I observed that:-
$g\left( 2 \right) =2$ $g\left( 4 \right) =4$ $g\left( 8 \right) =8$ $g\left( 16\right) =7$ $g\left( 32 \right) =5$ $g\left( 64 \right) =1$
then
$g\left( 128 \right) =2$ $g\left( 256\right) =4$ $g\left( 512 \right) =8$ $g\left( 1024 \right) =7$ $g\left( 2048 \right) =5$ $g\left( 4096 \right) =1$
and this pattern is repeating again and again.
$g\left( 8192 \right) =2$ $g\left( 16384 \right) =4$
and so on.........
So, The pattern for powers of 2 is $$2,4,8,7,5,1,2,4,8,7,5,1$$
and
The pattern for powers of 3 is $$3,9,9,9,9,9,9,9,9,9,9,9$$
The pattern for powers of 4 is $$4,7,1,4,7,1,4,7,1,4,7,1$$
The pattern for powers of 5 is $$5,7,8,4,2,1,5,7,8,4,2,..$$
Why does it work?
Does it work for powers of all integers?
Does it work in all bases?
Note by Archit Boobna
3 years, 3 months ago
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2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
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Hint: $$\bmod 9$$, divisibility rule.
- 3 years, 3 months ago
One line. That beautiful number $$9$$! +1
- 3 years, 3 months ago
Thanks so much!
I got it. The sum of digits mod 9 is same as the number mod 9.
And 2^n mod 9 is repeating every few terms.
- 3 years, 3 months ago | crawl-data/CC-MAIN-2018-34/segments/1534221210362.19/warc/CC-MAIN-20180815220136-20180816000136-00088.warc.gz | null |
## Lecture 5. Exponential of vector fields and solutions of differential equations
Let $x\in C^{1-var} ([0,T], \mathbb{R}^d)$ and let $V : \mathbb{R}^e \to \mathbb{R}^{e\times d}$ be a Lipschitz continuous map. In order to analyse the solution of the differential equation,
$y(t)=y_0+\int_0^t V(y(s)) dx(s),$
and make the geometry enter into the scene, it is convenient to see $V$ as a collection of vector fields $V=(V_1, \cdots, V_d)$, where the $V_i$‘s are the columns of the matrix $V$. The differential equation then of course writes
$y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s),$
Generally speaking, a vector field $V$ on $\mathbb{R}^{e}$ is a map
$\begin{array}{llll} V: & \mathbb{R}^{e}& \rightarrow & \mathbb{R}^{e} \\ & x & \rightarrow & (v_{1}(x),...,v_{e}(x)). \end{array}$
A vector field $V$ can be seen as a differential operator acting on differentiable functions $f: \mathbb{R}^{e} \rightarrow \mathbb{R}$ as follows:
$Vf(x)=\langle V(x), \nabla f (x) \rangle= \sum_{i=1}^e v_i (x) \frac{\partial f}{\partial x_i}.$
We note that $V$ is a derivation, that is for $f,g \in \mathcal{C}^{1} (\mathbb{R}^e , \mathbb{R} )$,
$V(fg)=(Vf)g +f (Vg).$
For this reason we often use the differential notation for vector fields and write:
$V=\sum_{i=1}^d v_i(x) \frac{\partial }{\partial x_i}.$
Using this action of vector fields on functions, the change of variable formula for solutions of differential equations takes a particularly concise form:
Proposition: Let $y$ be a solution of a differential equation that writes
$y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s),$
then for any $C^1$ function $f: \mathbb{R}^{e} \rightarrow \mathbb{R}$,
$f(y(t))=f(y_0)+\sum_{i=1}^d \int_0^t V_i f (y(s)) dx^i(s).$
Let $V$ be a Lipschitz vector field on $\mathbb{R}^e$. For any $y_0 \in \mathbb{R}^e$, the differential equation
$y(t)=y_0+\int_0^t V(y(s)) ds$
has a unique solution $y: \mathbb{R} \to \mathbb{R}^e$. By time homogeneity of the equation, the flow of this equation satisfies
$\pi ( t_1 , \pi( t_2 ,y_0 ) )=\pi (t_1 +t_2,y_0).$
and therefore $\{ \pi( t, \cdot), t \in \mathbb{R}\}$ is a one parameter group of diffeomorphisms $\mathbb{R}^e \to \mathbb{R}^e$. This group is generated by $V$ in the sense that for every $y_0 \in \mathbb{R}^e$,
$\lim_{t\to 0} \frac{\pi(t,y_0) -y_0}{t}=V(y_0).$
For these reasons, we write $\pi(t,y_0)=e^{tV}(y_0)$. Let us now assume that $V$ is a $C^1$ Lipschitz vector field on $\mathbb{R}^e$. If $\phi :\mathbb{R}^e \to \mathbb{R}^e$ is a diffeomorphism, the pull-back $\phi^{\ast}V$ of the vector field $V$ by the map $\phi$ is the vector field defined by the chain rule,
$\phi^{\ast}V (x)=(d \phi^{-1} )_{\phi (x) } \left( V (\phi(x)) \right)$. In particular, if $V'$ is another $C^1$ Lipschitz vector field on $\mathbb{R}^e$, then for every $t \in \mathbb{R}$, we have a vector field $(e^{tV})^{\ast} V'$. The Lie bracket $[V,V']$ between $V$ and $V'$ is then defined as
$[V,V']=\left( \frac{d}{dt} \right)_{t=0} (e^{tV})^{\ast}V'.$
It is computed that
$[ V, V' ](x)=\sum_{i=1}^e \left( \sum_{j=1}^e v_j (x) \frac{\partial v'_i}{\partial x_j}(x)- v'_j (x) \frac{\partial v_i}{\partial x_j}(x)\right)\frac{\partial}{\partial x_i}.$
Observe that the Lie bracket obviously satisfies $[V,V']=-[V',V]$ and the so-called Jacobi identity that is:
$[V,[V',V'']]+[V',[V'',V]]+[V'',[V,V']]=0.$
What the Lie bracket $[V,V']$ really quantifies is the lack of commutativity of the respective flows generated by $V$ and $V'$.
Lemma: Let $V,V'$ be two $C^1$ Lipschitz vector fields on $\mathbb{R}^e$. Then, $[V,V']=0$ if and only if for every $s,t \in \mathbb{R}$,
$e^{sV} e^{t V'}=e^{sV+tV'}=e^{t V'} e^{sV}.$
Proof: This is a classical result in differential geometry, so we only give one part the proof. From the very definition of the Lie bracket and the multiplicativity of the flow, we see that $[V,V']=0$ if and only if for every $s \in \mathbb{R}$, $(e^{sV})^{\ast}V'=V'$. Now, suppose that $[V,V']=0$. Let $y$ be the solution of the equation
$y(t)=y_0+\int_0^t V'(y(s)) ds.$
Since $(e^{sV})^{\ast}V'=V'$, we obtain that $e^{sV} (y(t))$ is also a solution of the equation. By uniqueness of solutions, we obtain that
$e^{sV}(y(t))=e^{tV'} ( e^{sV}(y_0)).$
As a conclusion,
$e^{sV} e^{t V'}=e^{t V'} e^{sV}$
$\square$
If we consider a differential equation
$y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s),$
as we will see it throughout this class, the Lie brackets $[V_i,V_j]$ play an important role in understanding the geometry of the set of solutions. The easiest result in that direction is the following:
Proposition: Let $x\in C^{1-var} ([0,T], \mathbb{R}^d)$ and let $V_1,\cdots, V_d$ be $C^1$ Lipschitz vector fields on $\mathbb{R}^e$. Assume that for every $1 \le i,j \le d$, $[V_i,V_j]=0$, then the solution of the differential equation
$y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s), \quad 0 \le t \le T,$
can be represented as
$y(t)= \exp \left( \sum_{i=1}^d x^i(t) V_i \right) (y_0).$
Proof: Let
$F(x_1,\cdots,x_n)= \exp \left( \sum_{i=1}^d x_i V_i \right) (y_0).$
Since the flows generated by the $V_i$‘s are commuting, we get that
$\frac{\partial F}{\partial x_i}(x)=V_i (F(x)).$
The change of variable formula for bounded variation paths implies then that $F(x^1(t),\cdots,x^n(t))$ is a solution and we conclude by uniqueness $\square$
This entry was posted in Rough paths theory. Bookmark the permalink.
### 5 Responses to Lecture 5. Exponential of vector fields and solutions of differential equations
1. taramata says:
It seems I can not follow your arugments when you define the pull-back of a vector field. On the RHS you have a differential and some function in the index. Is this well-known notation?
2. The notation $(d \phi^{-1} )_{\phi (x) } \left( V (\phi(x)) \right)$ means the following: $(d \phi^{-1} )_{\phi (x) }$ is the differential of $\phi^{-1}$ at the point $\phi(x)$. This is a linear map that we apply then to the vector $V (\phi(x))$.
3. taramata says:
Thank you for the response. As far as I understand $(d \phi^{-1})_{\phi(x)}(V(\phi(x)))=D^{-1}\phi(x) V(x)$, here $D^{-1} \phi(x)$ stands for the inverse of the derivative of $\phi$, in our case this is some matrix. Right? | crawl-data/CC-MAIN-2018-13/segments/1521257647244.44/warc/CC-MAIN-20180319234034-20180320014034-00519.warc.gz | null |
Life on Earth probably started some 3 billion years ago, give or take several hundred million years. But multicellular organisms, including animals, didn't start to show up until much more recently, within the last 700 million years.
Paleontologists have unearthed an extraordinary diversity of fossils from the period starting about 540 million years ago, a period known as the Cambrian explosion. But fossils from before this time are scant or peculiar, making it difficult to pinpoint what type of creature was the first in the animal line.
But new genetic and paleontological evidence assembled by MIT researchers may finally offer an answer.
So what was Earth's first animal? It turns out, it was probably a simple sea sponge, reports Phys.org.
The clues didn't come from fossils in the traditional sense, but rather from traces of certain molecules found in ancient rocks — molecular fossils, if you will. Basically, when an animal dies and decays, it leaves evidence of its existence in the form of biomarkers and chemicals, even when it doesn't fossilize. So, theoretically, scientists can study ancient rocks and look for biosignatures unique to certain kinds of animals, even in the absence of fossils.
Previous research in 1994 had identified one chemical in particular, 24-isopropylcholestane (or 24-ipc for short), in high amounts in Cambrian and slightly older rocks. This substance is a lipid molecule, or sterol, a modified version of cholesterol, and it's known to be produced by sponges and a few other organisms alive today.
Then, in 2009, another research team confirmed the presence of 24-ipc in 640-million-year-old rock samples from Oman. The sheer age of these rock samples means they could very well represent traces of the first animals to evolve on Earth.
To assemble the puzzle of what kinds of animals might have produced this 24-ipc, researchers turned to genetic analysis. They surmised that if they could identify the gene responsible for making 24-ipc and find the organisms that carry this gene, they could trace when the gene evolved in those organisms.
The gene they identified, it turns out, is found in just the right form in both sponges and some types of algae. Researchers then performed genetic analysis to determine whether sea sponges or algae had evolved this gene first. The results were definitive: it was the sponges. Even more telling, the genetic analysis revealed a rough date for when the gene likely first appeared among sponges: 640 million years ago.
The pieces of the puzzle fit together perfectly, offering a compelling case that the sponge — or at least some manifestation of a sponge-like creature — was the first animal to inhabit the planet.
Think about that the next time you're scrubbing yourself down in the shower.
"This goes to show how much we still don't know about early animal life, how many discoveries there are left, and how useful, when done properly, these molecular fossils can be to help fill in those gaps," said David Gold, a postdoc in MIT's Department of Earth, Atmospheric and Planetary Sciences.
The results have been published in the Proceedings of the National Academy of Sciences. Gold is the lead author on the paper, along with senior author and EAPS Professor Roger Summons. | <urn:uuid:4d1fd9ad-51f2-4e55-b137-2cb2f6d6b8f4> | {
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Richard Owen created the Natural History Museum, and was the first superintendent. The year the Museum opened, this portrait was painted by William Holman Hunt, one of the most significant artists of his time.
Drawing of a human skeleton (left) and a gorilla skeleton by Richard Owen, 1866.
Richard Owen's teachers would not have predicted that the schoolboy prankster would become the eminent scientist shown in this portrait. But while training to be a surgeon he was taught to compare the body structures of humans with other animals, and his attention finally focused. He had found the vocation that would drive him to the top of the scientific ranks.
Owen led the science of comparative anatomy for almost 60 years, naming and describing hundreds of species.
He rose to fame in 1832 by publishing Memoir on the Pearly Nautilus, which is still in print today.
Owen was hugely respected by his peers, but also feared and resented. He guarded his professional reputation aggressively, challenging anyone who stood in his way. He clashed with Charles Darwin over opposing theories of evolution, and their friendship never recovered.
In contrast, Owen’s family relationships were warm and loving.
Design for the main entrance of the Natural History Museum, sketched by architect Alfred Waterhouse in about 1872. © RIBA Library Drawings Collection
In 1856, the role of superintendent of natural history was created for Owen at the British Museum.
By now the most powerful scientist in the country, he lobbied for the natural history specimens to be moved to a separate museum.
Owen was determined to create a building that would do justice to the amazing diversity of nature and enable both scientists and the public to learn more about the national collections.
The first stones of our building at South Kensington in London were laid in 1873. Owen's ambitious plan was finally realised when this museum opened to the public in 1881.
That same year, William Holman Hunt painted Owen's portrait. A founding member of the Pre-Raphaelite Brotherhood, Hunt was one of the most famous artists of his day.
The Pre-Raphaelites' vibrant new approach to art reflected the dynamic and changing Victorian society. They painted often controversial social, religious and natural themes. The portrait of Owen is 1 of only 3 known portraits by Hunt - a fitting tribute to a remarkable, complex man. | <urn:uuid:e55b64a5-3ecf-45da-96ce-7cf26d219e9d> | {
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C3 plants involve direct carbon fixation of CO2. That is, the initial steps involve the CO2 being bound to ribulose bisphosphate to produce two molecules of three-carbon compound (i.e. 3-phosphogylycerate). The key enzyme that catalyzes carbon fixation is rubisco.
C3 plants must however be in areas where CO2 concentration is high, temperature and light intensity are moderate, and ground water is abundant. This is because in hot areas, the stomata are closed to prevent water loss. However, it results in the rise of O2 level. When this occurs, rubisco reacts with O2 instead of CO2, and leads to photorespiration, which in turn, causes wasteful loss of CO2 in C3 plants. | <urn:uuid:24397a2e-bc87-4fd9-8a0f-1de9be2ce662> | {
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# T4T Is it Odd or Even?
Click to access fully formatted lesson and materials:
Lesson excerpt:
Understand the meaning of odd and even numbers
Theoretical Foundation: As students in second grade begin to look at the properties and attributes of numbers, they begin to have an understanding of odd and even. It is not enough for students to just be able to identify an odd or an even number, students should build a conceptual understanding of why a number is classified as odd or even.
Estimated Time: 40 minutes
Materials: 2 dice per student (or pair of students), 1-6 or 0-9 counters, recording and sorting pages, Even Steven and Odd Todd, by Kathryn Cristaldi
Description:
1. Introduce the vocabulary “odd” and “even”.
2. Read and discuss the book Even Steven and Odd Todd. As you read the book, draw or model the numbers of items that are modeled in the story. As you do this, be sure to put them into 2 equal groups and a left over (if odd).
3. After reading the story, have students identify different times in the story in which an item was odd or was even. Be sure to have students explain their reasoning.
4. Give each student (or pair of students) two dice and at least 12 (18 counters if using the 0-9 dice) counters.
5. Have students roll the dice and count out the number of counters that are shown on the dice.
6. Once students have the correct number of counters counted out, they are to record the number of counters on their recording sheet.
7. Next, have the students separate the counters into two equal groups (on the recording sheet) until all counters are in the circles or until there is only one left over.
8. Have the students decide whether or not the number on the dice is odd or even and record their findings on their recording sheet.
9. Continue this for 12 turns.
10. Create a class chart, with numbers 1-12, have students to help fill in the odd/even column.
Differentiation Suggestions:
1. Use 6 dice so the numbers can go up to 36. This will allow students to begin to see the patterns in the digits that make up odd/even numbers.
2. Allow students to work with a partner. Partners can take turns rolling dice, counting out counters and recording.
3. Break this lesson into two smaller lessons.
Probing Questions:
1. Do you notice any patterns with the numbers that are odd or even?
2. What happens every time you add one to the number?
3. What happens when you take one (or two) away from the number that you rolled? Is it still even/odd? Does that happen every time?
Assessment:
1. Does the student see any patterns to the way odd and even numbers are classified?
2. Do the students understand why numbers are classified as odd or even? | crawl-data/CC-MAIN-2023-50/segments/1700679100873.6/warc/CC-MAIN-20231209071722-20231209101722-00818.warc.gz | null |
Today’s activity was designed to target the surface area & volume expectation in my MFM2P course. I’ve done this one a few times in the past.
This time around I started by asking them to guess whether or not these two boxes had the same volume (I told them they both hold 12 cans – which is also written on the box):Over half of my students said NO – they were not the same volume. I sent them to their boards to check whether or not they were right. The volumes turned out not to be exactly the same, but we discussed that if we measured in cans, they both had the same volume; 12 cans. But if we measure in square centimetres, one had slightly more volume.
“But why isn’t our answer for the Pepsi box the same as that other group? . . . Oh, we must have measured differently.”
So this spurred a quick discussion of being accurate in our measurements.
The group working on the whiteboard pictured above used the formula from their formula sheet to calculate the surface area of the box. This group and others had initially misinterpreted the formula, adding instead of multiplying dimensions, etc. I called groups back to their boards, discussed how to “read” the formulas & asked them to revise their work.
One of our 5 groups tried to solve by calculating the area of each face of the box:You can see their volume work from earlier at the top of their board. their surface area work is messy but towards the middle you can see them calculating the length x width of each rectangle. On the left they are multiplying those answers by 2. It doesn’t look like they got to the point of adding them together.
I called attention to the 2 different methods used by the class; surface area formulas VS summing the areas of the faces (working with nets).
The rest of class time was spent working on the homework:
Surface area using nets on KhanAcademy
or Surface area (for the 4 students that have already mastered the previous exercise).
I circulated helping students get started on their homework.
In Dan Meyer‘s 3 act math, the 3rd act is checking if we are correct somehow. Lately, the 3rd act in my class has been more about the metacognitive task of discussing their various strategies in solving the problem. Does that make the activity less powerful if we don’t physically check if we modelled correctly after? Perhaps I need to create the act 3 as a photo or video where I lay each box out flat as a net & show the surface area of each. Or should I cut them up to rearrange them into similar shapes to get the visual impact of which one has a larger surface area?
I missed the boat today on having my students generate questions we could solve for this scenario. I should have had a slide in my Pear Deck slideshow at the start asking what Mathematical questions we could ask about these two boxes:
Next time I’ll add that in.
All the materials for this activity are here.
Update (2015.11.10): Last night I bought another 2 drink boxes so that I could cut them up, measure carefully & calculate the surface areas. So at the start of class today (the day after the original activity) we looked at all of our solutions from the previous day to see which group best modelled the correct surface area:
– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON) | <urn:uuid:181c59e3-e5d4-4ca6-bdd5-3a37dc981c90> | {
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Schmidt telescope, also called Schmidt camera, telescope in which a spherical primary mirror receives light that has passed through a thin aspherical lens, called a correcting plate, that compensates for the image distortions—namely, spherical aberrations—produced by the mirror. The Schmidt telescope is thus a catadioptric telescope; i.e., its optics involve both the reflection and refraction of light. Because the Schmidt telescope uses a spherical collecting mirror instead of a paraboloidal one (as conventional reflecting telescopes do), it is free from astigmatism and so has a wide field of view. The Schmidt instrument can, in effect, provide a sharper image of a larger area of the celestial sphere than ordinary reflectors and is thus ideal for star surveys.
The device was invented in 1930 by optician Bernhard Schmidt of the Bergedorf Observatory in Hamburg. The Schmidt-Maksutov telescope, invented by Russian optician Dmitry D. Maksutov in 1941, is similar in design and purpose to the Schmidt telescope but has a spherical meniscus, a lens in which one side is concave and the other is convex, in place of the correcting plate of the Schmidt. | <urn:uuid:05cf7612-bf8d-460c-8efd-3fa17d557937> | {
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What the Terms Revaluation and Devaluation Mean
The terms revaluation and devaluation are used instead of appreciation and depreciation, respectively. When you read any financial newspaper, you note that a weakening in the dollar is reported as depreciation of the dollar. But if the same happens to the Chinese yuan, it’s phrased as the yuan being devalued. Professional people use a certain language, and it’s important to understand and use this language.
However, to use the terms revaluation or devaluation, you need substantial government interventions in the exchange rate. The Chinese yuan is a good example for a currency whose value with respect to other currencies is determined by the Chinese government. The goal is to identify devaluation and revaluation, for example, on a graph.
The figure shows the yuan–dollar annual exchange rate for the period 1981–2011. The yuan–dollar exchange rate was CNY1.71 per dollar in 1981 and then steadily increased until 1994, when it reached CNY8.6397. Between 1994 and 2005, the yuan–dollar exchange rate remained above CNY8 per dollar.
Beginning in 2005, the exchange rate declined, from CNY8.1936 in 2005 to CNY6.4630 in 2011. In the graph, you can use this information to calculate the percent change in the Chinese yuan and apply your knowledge of the terminology associated with the change in the exchange rate.
Consider the period of 1981–1994:
Here, E stands for the yuan–dollar exchange rate. A 405 percent increase in the yuan–dollar exchange rate is shown over a period of 14 years, which translates into an average annual devaluation of the yuan of almost 29 percent (405 / 14).
Now apply the percent change formula to the 2005–2011 period:
This result indicates an average annual decline in the number of yuan necessary to buy one dollar of about 3.02 percent in the past seven years (21.12 / 7), which is called a revaluation. | <urn:uuid:d0178db7-1074-4e3a-bb8d-e05114075425> | {
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There remains a widespread sentiment that the Indigenous populations of the Awarak, Taino, and Lucayan from which Italian explorer Christopher Columbus discovered in 1492 was, by many accounts, “barbaric savages.” Childlike, violent, godless, deficient in government, technology, philosophy, and other distinctive features that at the time was viewed as necessary elements in the formation of an authentic civilization.
Conversely, Columbus has often been, until most recently, popularly understood as the representative of a civilized Eurocentric worldview whose advancements in the areas of government, technology, and philosophy, amongst other societal functions, transformed him into a bearer of development that elevated the statuses of the “inferior” indigenous populations. Future settlers and colonizers of the Americas would also perceive and widely perpetuate this construct of American history.
The real irony is that long before Columbus made contact with the “New World,” – a world that he believed was in India – many Native American populations had already developed highly advanced civilizations. Some of these Native American cultures had developed extensive irrigation networks that contended with those found in Europe. They had formulated an acute understanding of nature. They had domesticated crops. They had incredible knowledge surrounding natural medicines. They had also developed extraordinarily sophisticated forms of governments from which the concept of freedom was not only treasured but was also the leading principle throughout their respective administrative bodies. However, these high achievements from America’s indigenous peoples are seldom acknowledged as significant foundational elements of the American cultural landscape and beyond.
The real irony is that long before Columbus made contact with the “New World,” – a world that he believed was in India – many Native American populations had already developed highly advanced civilizations.
The scope of discussion regarding American history has often perpetuated the mythology mentioned above of Eurocentrism, diminishing much if not all of the impact that America’s indigenous peoples have had in shaping the historical trajectory of the continent. It has only been in the last 50 years that the prevailing thoughts of history had shifted, with prior mainstream historians interpreting everything that was notable about America occurring in and after 1492, with the origins of America residing in Europeans bringing civilization to the traditional inhabitants and taming the natural environment in the process.
Notions contributing to the roots of Manifest Destiny would also come into play as well, and with it, the search for the perfect civilization with a divine will serving to propel America towards greatness. This approach and understanding derived from applications of Western thought at the time, a thought process which viewed the Western “civilized” world as being at the forefront from which all other human societies were to admire and follow – usually at the behest of applied force along the way.
Historians who had subscribed to this Eurocentric interpretation of history have strengthened it by looking at the introduction of Europeans onto the American continent as a continual process of improvement, whether it be the traditional inhabitants or natural environment. Indeed, many of the first settlers had viewed it in the same fashion. Historians who follow this ideology have frequently selected – and simply disregarded – valuable information regarding the truthful past surrounding the interactions between indigenous peoples and European settlers. This doctrine has manifested itself in a variety of destructive ways, including the consideration that the persecution of the native populations of America was an inevitable but fundamentally positive outcome.
Historians who follow this ideology have frequently selected – and simply disregarded – valuable information regarding the truthful past surrounding the interactions between indigenous peoples and European settlers.
This basis for understanding is undoubtedly dangerous thinking. For any civilization to believe that they hold ultimate knowledge and that others need to be given it, inexplicable and horrible actions can be pursued to administrate that perceived superior knowledge. History has shown us this to be true. In the case of the Americas, the notion of civilizing the native savages has persisted within Eurocentric views of American history. This understanding has often come in the form rationalizing many of the destructive practices exerted upon the indigenous populations, in turn envisioning a world where the American Indian interpreted as the “other,” a term used to invoke the cultural inferiority of one group to that of another.
The origins of America, and as most Americans understand it, consists of the immense journey of the European settlers to the Americas, where they transformed it into new expressions of Europe and brought along with it the cultures of the Old World. While this is true, it completely disregards the fact that European settlers and colonizers did not land on virgin soil in American but rather in the midst of a multitude of complex and rich indigenous peoples and civilizations.
The story of Europeans settlers and colonizers is not one of a foreign people exploring an unoccupied continent; it is one of human interaction and immense struggle that equaled a fundamental clash of worldviews. The discounting of this facet of history is not only harmful to contributions of American Indians and other indigenous populations throughout the Americas but on a deeper level, who we are as a collective nation.
Miguel Douglas is the executive director of American Indian Republic and is an enrolled member of the Puyallup Tribe of Indians. He has written extensively on Indian gaming and its effects on American Indian communities. He has received a Master’s Degree in Interdisciplinary Studies from the University of Washington. | <urn:uuid:76d32bb3-b539-417a-9c56-b158e1d2d726> | {
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# A Comprehensive Guide to Converting Cubic Meters to Cubic Yards
Cubic Meters:
Output: `Press calculate`
## Understanding the Conversion from Cubic Meters to Cubic Yards
Whether you're managing a construction project, dealing with landscaping, or simply curious about measurements, understanding how to convert cubic meters to cubic yards is essential. These measurements often arise in various fields, but translating between them can seem daunting. Let’s break down this topic and make it simple and manageable.
## Decoding the Basics: What are Cubic Meters and Cubic Yards?
Cubic Meters (m³): This is the metric unit for volume. One cubic meter is equivalent to the volume of a cube with edges of one meter each. This unit is predominant in countries that use the metric system and is commonly used in fields like physics, engineering, and everyday measurements.
Cubic Yards (yd³): This is an imperial unit for volume. One cubic yard is the volume of a cube with edges of one yard (3 feet) each. This measurement is widely used in the United States and the United Kingdom, especially in construction, landscaping, and other industries.
## The Conversion Formula
Formula to Convert Cubic Meters to Cubic Yards:` yd³ = m³ * 1.30795 `
To convert cubic meters to cubic yards, you simply multiply the volume in cubic meters by 1.30795. This constant comes from the fact that one cubic meter equals approximately 1.30795 cubic yards.
## Real-life Application of the Conversion
Example 1: Planning a Garden
Imagine you're planning to fill a garden bed with soil. If you have the soil measured in cubic meters but your project plan specifies the volume in cubic yards, you’ll need this conversion.
Let’s say you have 5 cubic meters of soil. Using the formula:
Calculation: 5 m³ * 1.30795 = 6.53975 yd³
So, you’ll need approximately 6.54 cubic yards of soil.
## Frequently Asked Questions
### Q: Why is it important to understand cubic meters to cubic yards conversion?
A: Both units are used in different regions and across various industries. Understanding the conversion helps in international projects, scientific calculations, and practical applications like construction and landscaping.
### Q: Can I use a calculator for this conversion?
A: Absolutely! There are numerous online calculators and apps available. However, knowing the formula (cubic meters * 1.30795) helps in understanding the process and can be useful in situations where you cannot access a calculator.
### Q: Are there other conversions I should be aware of?
A: Yes. Depending on your field, you might also need to convert cubic meters to cubic feet, liters, or gallons. Each conversion uses a different constant.
## Summary
Converting between cubic meters and cubic yards is a fundamental skill in various fields, from construction to everyday planning. By understanding the formula and its real-life applications, you can tackle any project involving these measurements with confidence. Whether it's filling a garden bed, planning construction materials, or engaging in scientific research, you're now equipped to navigate these conversions effectively.
Tags: Conversion, Geometry, Volume | crawl-data/CC-MAIN-2024-30/segments/1720763514494.35/warc/CC-MAIN-20240713114822-20240713144822-00615.warc.gz | null |
For years, wearables have had something of a power conundrum. Without a bulky and annoying rechargeable battery, there's no way to supply components with the energy they need to operate.
Multiple research teams across the globe are tackling this problem - trying to figure out how to harvest energy from the body's motion or its surroundings. But to date, these haven't succeeded in producing enough power, or they aren't stretchy and flexible enough to conform to the human body.
But now engineers at the University of California San Diego claim to have made a breakthrough. They've developed a biofuel cell that can extract enough energy from the body's sweat to power electronics like LEDs and Bluetooth radios.
How it works
Using a combination of chemistry, advanced materials and electronic interfaces, their fuel cell delivers ten times more power per surface area than any existing wearable biofuel cell.
Here's how it works. Using lithography, the team built up a stretchy electronic foundation out of gold, then screen printed three-dimensional carbon nanotube-based cathode and anode arrays on top. Finally, they filled the cell with an enzyme that oxidises the lactic acid found in human sweat to generate a current.
The challenge came in increasing the cell's energy density. "We needed to figure out the best combination of materials to use and in what ratio to use them," Amay Bandodkar, one of the first authors on a describing the technology, published in Energy & Environmental Science
In tests, the team hooked up the cell to a custom made circuit board and had a team of volunteers cycle on a stationary bike. They were able to power a blue LED for about four minutes.
The team says that it's hoping to improve that lifespan by figuring out a way to store the energy produced and then release it gradually. They'd also like to replace the silver oxide used in the cathode, which degrades over time, with something more stable. | <urn:uuid:0e28b176-970a-43f6-9b8a-c92754ed2efd> | {
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Rounding and estimation
This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.
Free course
# 1.4.1 Try some yourself
## Activity 5
Round a measurement of 1.059 metres:
• (a) to the nearest whole number of metres;
• (b) to two decimal places;
• (a) 1.059 m rounded to the nearest whole number of metres is 1 m.
• (b) 1.059 m rounded to two decimal places is 1.06 m.
## Activity 6
Round each of the numbers below to:
• (a) to 1 d.p.
• (b) to 2 d.p.
• (c) to 3 d.p.
0.472 65, 13.959 943, 56.098 27
• (a) 0.5, 14.0, 56.1
• (b) 0.47, 13.96, 56.10
• (c) 0.473, 13.960, 56.098
## Activity 7
The table below contains some errors. Identify them and write down the correct rounding.
Rounded to
Number 1 d.p. 2 d.p. 3 d.p.
3.141 592 6 3.10 3.14 3.150
22/7 3.1 3.04 3.142
0.019 999 0.0 0.19 0.200
The corrected entries are in bold.
Rounded to
Number 1 d.p. 2 d.p. 3 d.p.
3.141 592 6 3.1 3.14 3.142
22/7 3.1 3.14 3.143
0.019 999 0.0 0.02 0.020
## Activity 8
Imagine you are alone on an island and that your only source of drinking water is a wrecked ship's full water tank. It measures 4 metres long by 3.45 metres wide and is 2.84 metres high. Use reasonable rounded values to make an estimate (without using your calculator) of how long you can survive until it rains, assuming that you need 2 litres per day and that the water remains perfectly fresh. You may need the conversion 1 cubic metre contains 1000 litres. (Note that volume = length × width × height.)
Rounding to approximate accuracy and then calculating the volume of the tank by using the formula length × width × height gives a volume of approximately 4 × 3 × 3 = 36 cubic metres. 1 cubic metre contains 1000 litres, so 36 cubic metres contains 36 × 1000 = 36 000 litres.
If you need 2 litres per day of water then you can survive 36 000 ÷ 2 = 18 000 days before it rains. This is about 20 000 days and taking a year to be about 400 days means that you can survive about 20 000 ÷ 400 = 50 years.
The accurate answer would give 53.69 years (to 2 d.p.). You might not be too concerned to be told that you had only 50 years rather than 53.69 years to live on your deserted island if it did not rain!
MU120_4M2 | crawl-data/CC-MAIN-2019-09/segments/1550249556231.85/warc/CC-MAIN-20190223223440-20190224005440-00601.warc.gz | null |
lect8 - Todays topics The notion of a relation properties...
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1 The notion of a relation properties of relations on a set Today’s topics: Relations A “relation” is a fundamental mathematical notion expressing a relationship between sets It’s an abstract notion useful for modeling many different relationships
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2 Example . Let S be set of UCF students, and C be a set of classes. Then we can consider the relation “is taking class” from S to C S C x y This relation can be described by the set of pairs: “is taking class”={( x , y )| x S , y C and student x is taking class y }
3 More examples of relations: “parent-of” “child-of” “likes” “meet one another today” “less then” = {( a , b ) | a , b A and a < b } where A ={1, 2, …5} “less then” = { (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)}
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4 Let A = {1, 2, 3, 4}. Which ordered pairs are in the relation R = {( a, b ) | a divides b }? R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)} A 1 2 3 4 A 1 2 3 4
5 Consider the following relations on the set of integers Z: R 1 ={( a , b ) | a b } R 2 ={( a , b ) | a > b } R 3 ={( a , b ) | a = b or a = - b } R 4 ={( a , b ) | a = b +1 } R 5 ={( a , b ) | a + b 3 } Which of these relations contain each of the pairs (1, 1), (1, 2), (2, 1), (1, - 1) and (2, 2)?
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6 equal” = {( a , b )| a , b Power({1, 2, 3}) and | a| =| b | } “subset” ={( a , b )| a , b Power({1, 2, 3}) and a b } equal” = {( , ), ({1}, {1}), ({1}, {2}), ({2}, {1}), … ({1, 2}, {2, 3}), ({2, 3}, {1, 2}), …({1, 2, 3}, {1, 2, 3})} If R is set of real numbers, R × R is set of points ( x , y ) in plane. “circle”={( x , y )| x , y R and x 2 + y 2 =1} subset” = {( , ), ( , {1}), ( , {2}), . .. ( , {1, 2, 3}), ({1}, {1}), ({1}, {1, 2}), ({1}, {1, 2, 3}), ({2}, {2}), … ({1, 2}, {1, 2}), ({1, 2}, {1, 2, 3}), ({1, 3}, {1, 3}), ({1, 3}, {1, 2, 3}), ({2, 3}, {1, 2, 3}), …({1, 2, 3}, {1, 2, 3})}
7 You don’t need to give a meaningful name to a relation. The only thing that really matters about relations is that we know which elements in A are related to which element of B. A relation R is completely described if we know R -related pairs Suppose A ={1, 2, 3}, B ={ r , s } and we know 1 Rr , 2 Rs , 3 Rr , then we know everything we need to know about R.
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lect8 - Todays topics The notion of a relation properties...
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Ask a homework question - tutors are online | crawl-data/CC-MAIN-2017-22/segments/1495463607802.75/warc/CC-MAIN-20170524055048-20170524075048-00392.warc.gz | null |
# Polynomials in Standard Form
## Understand polynomials as specific kinds of Algebraic expressions
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Polynomials in Standard Form
### Polynomials in Standard Form
So far we’ve seen functions described by straight lines (linear functions) and functions where the variable appeared in the exponent (exponential functions). In this section we’ll introduce polynomial functions. A polynomial is made up of different terms that contain positive integer powers of the variables. Here is an example of a polynomial:
\begin{align*}4x^3+2x^2-3x+1\end{align*}
Each part of the polynomial that is added or subtracted is called a term of the polynomial. The example above is a polynomial with four terms.
The numbers appearing in each term in front of the variable are called the coefficients. The number appearing all by itself without a variable is called a constant.
In this case the coefficient of \begin{align*}x^3\end{align*} is 4, the coefficient of \begin{align*}x^2\end{align*} is 2, the coefficient of \begin{align*}x\end{align*} is -3 and the constant is 1.
Degrees of Polynomials and Standard Form
Each term in the polynomial has a different degree. The degree of the term is the power of the variable in that term.
\begin{align*}& 4x^3 && \text{has degree} \ 3 \ \text{and is called a cubic term or} \ 3^{rd} \ \text{order term}.\\ & 2x^2 && \text{has degree} \ 2 \ \text{and is called a quadratic term or} \ 2^{nd} \ \text{order term}.\\ & -3x && \text{has degree} \ 1 \ \text{and is called a linear term or} \ 1^{st} \ \text{order term}.\\ & 1 && \text{has degree} \ 0 \ \text{and is called the constant}.\end{align*}
By definition, the degree of the polynomial is the same as the degree of the term with the highest degree. This example is a polynomial of degree 3, which is also called a “cubic” polynomial. (Why do you think it is called a cubic?).
Polynomials can have more than one variable. Here is another example of a polynomial:
\begin{align*}t^4-6s^3t^2-12st+4s^4-5\end{align*}
This is a polynomial because all the exponents on the variables are positive integers. This polynomial has five terms. Let’s look at each term more closely.
Note: The degree of a term is the sum of the powers on each variable in the term. In other words, the degree of each term is the number of variables that are multiplied together in that term, whether those variables are the same or different.
\begin{align*}& t^4 && \text{has a degree of} \ 4, \ \text{so it's a} \ 4^{th} \ \text{order term}\\ & -6s^3t^2 && \text{has a degree of} \ 5, \ \text{so it's a} \ 5^{th} \ \text{order term}.\\ & -12st && \text{has a degree of} \ 2, \ \text{so it's a} \ 2^{nd} \ \text{order term}.\\ & 4s^4 && \text{has a degree of} \ 4, \ \text{so it's a} \ 4^{th} \ \text{order term}.\\ & -5 && \text{is a constant, so its degree is} \ 0.\end{align*}
Since the highest degree of a term in this polynomial is 5, then this is polynomial of degree \begin{align*}5^{th}\end{align*} or a \begin{align*}5^{th}\end{align*} order polynomial.
A polynomial that has only one term has a special name. It is called a monomial (mono means one). A monomial can be a constant, a variable, or a product of a constant and one or more variables. You can see that each term in a polynomial is a monomial, so a polynomial is just the sum of several monomials. Here are some examples of monomials:
\begin{align*}b^2 \qquad -2ab^2 \qquad 8 \qquad \frac{1}{4}x^4 \qquad -29xy\end{align*}
#### Identifying Constants and the Degree of a Polynomial
For the following polynomials, identify the coefficient of each term, the constant, the degree of each term and the degree of the polynomial.
a) \begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}
\begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}
The coefficients of each term in order are 1, -3, 4, and -5 and the constant is 7.
The degrees of each term are 5, 3, 2, 1, and 0. Therefore the degree of the polynomial is 5.
b) \begin{align*}x^4-3x^3y^2+8x-12\end{align*}
\begin{align*}x^4-3x^3y^2+8x-12\end{align*}
The coefficients of each term in order are 1, -3, and 8 and the constant is -12.
The degrees of each term are 4, 5, 1, and 0. Therefore the degree of the polynomial is 5.
#### Identifying Polynomials
Identify the following expressions as polynomials or non-polynomials.
a) \begin{align*}5x^5-2x\end{align*}
This is a polynomial.
b) \begin{align*}3x^2-2x^{-2}\end{align*}
This is not a polynomial because it has a negative exponent.
c) \begin{align*}x\sqrt{x}-1\end{align*}
This is not a polynomial because it has a radical.
d) \begin{align*}\frac{5}{x^3+1}\end{align*}
This is not a polynomial because the power of \begin{align*}x\end{align*} appears in the denominator of a fraction (and there is no way to rewrite it so that it does not).
e) \begin{align*}4x^\frac{1}{3}\end{align*}
This is not a polynomial because it has a fractional exponent.
f) \begin{align*}4xy^2-2x^2y-3+y^3-3x^3\end{align*}
This is a polynomial.
Often, we arrange the terms in a polynomial in order of decreasing power. This is called standard form.
The following polynomials are in standard form:
\begin{align*}4x^4-3x^3+2x^2-x+1\end{align*}
\begin{align*}a^4b^3-2a^3b^3+3a^4b-5ab^2+2\end{align*}
The first term of a polynomial in standard form is called the leading term, and the coefficient of the leading term is called the leading coefficient.
The first polynomial above has the leading term \begin{align*}4x^4,\end{align*} and the leading coefficient is 4.
The second polynomial above has the leading term \begin{align*}a^4b^3,\end{align*} and the leading coefficient is 1.
#### Writing Polynomials in Standard Form
Rearrange the terms in the following polynomials so that they are in standard form. Indicate the leading term and leading coefficient of each polynomial.
a) \begin{align*}7-3x^3+4x\end{align*}
\begin{align*}7-3x^3+4x\end{align*} becomes \begin{align*}-3x^3+4x+7\end{align*}. Leading term is \begin{align*}-3x^3\end{align*}; leading coefficient is -3.
b) \begin{align*}ab-a^3+2b\end{align*}
\begin{align*}ab-a^3+2b\end{align*} becomes \begin{align*}-a^3+ab+2b\end{align*}. Leading term is \begin{align*}-a^3\end{align*}; leading coefficient is -1.
c) \begin{align*}-4b+4+b^2\end{align*}
\begin{align*}-4b+4+b^2\end{align*} becomes \begin{align*}b^2-4b+4\end{align*}. Leading term is \begin{align*}b^2\end{align*}; leading coefficient is 1.
#### Simplifying Polynomials
A polynomial is simplified if it has no terms that are alike. Like terms are terms in the polynomial that have the same variable(s) with the same exponents, whether they have the same or different coefficients.
For example, \begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms, but \begin{align*}6x^2y\end{align*} and \begin{align*}6xy^2\end{align*} are not like terms.
When a polynomial has like terms, we can simplify it by combining those terms.
\begin{align*}& x^2+\underline{6xy} - \underline{4xy} + y^2\\ & \qquad \nearrow \qquad \nwarrow\\ & \qquad \text{Like terms}\end{align*}
We can simplify this polynomial by combining the like terms \begin{align*}6xy\end{align*} and \begin{align*}-4xy\end{align*} into \begin{align*}(6-4)xy\end{align*}, or \begin{align*}2xy\end{align*}. The new polynomial is \begin{align*}x^2+2xy+y^2\end{align*}.
Simplify the following polynomials by collecting like terms and combining them.
a) \begin{align*}2x -4x^2+6+x^2-4+4x\end{align*}
Rearrange the terms so that like terms are grouped together: \begin{align*}(-4x^2+x^2)+(2x+4x)+(6-4)\end{align*}
Combine each set of like terms: \begin{align*}-3x^2+6x+2\end{align*}
b) \begin{align*}2x -4x^2+6+x^2-4+4x\end{align*}
Rearrange the terms so that like terms are grouped together: \begin{align*}(a^3b^3-a^3b^3)+(-5ab^4+3ab^4)+2a^3b-a^2b\end{align*}
Combine each set of like terms: \begin{align*}0-2ab^4+2a^3b-a^2b=-2ab^4+2a^3b-a^2b\end{align*}
### Example
#### Example 1
Simplify and rewrite the following polynomial in standard form. State the degree of the polynomial.
\begin{align*}16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}
Start by simplifying by combining like terms:
\begin{align*} 16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}
is equal to
\begin{align*}(16x^2y^3-7x^2y^3)-3xy^5+(-2x^3y^2+2x^3y^2)+2xy\end{align*}
which simplifies to
\begin{align*}9x^2y^3-3xy^5+2xy.\end{align*}
In order to rewrite in standard form, we need to determine the degree of each term. The first term has a degree of \begin{align*}2+3=5\end{align*}, the second term has a degree of \begin{align*}1+5=6\end{align*}, and the last term has a degree of \begin{align*}1+1=2\end{align*}. We will rewrite the terms in order from largest degree to smallest degree:
\begin{align*}-3xy^5+9x^2y^3+2xy\end{align*}
The degree of a polynomial is the largest degree of all the terms. In this case that is 6.
### Review
Indicate whether each expression is a polynomial.
1. \begin{align*}x^2+3x^{\frac{1}{2}}\end{align*}
2. \begin{align*}\frac{1}{3}x^2y-9y^2\end{align*}
3. \begin{align*}3x^{-3}\end{align*}
4. \begin{align*}\frac{2}{3}t^2-\frac{1}{t^2}\end{align*}
5. \begin{align*}\sqrt{x}-2x\end{align*}
6. \begin{align*}\left ( x^\frac{3}{2} \right )^2\end{align*}
Express each polynomial in standard form. Give the degree of each polynomial.
1. \begin{align*}3-2x\end{align*}
2. \begin{align*}8-4x+3x^3\end{align*}
3. \begin{align*}-5+2x-5x^2+8x^3\end{align*}
4. \begin{align*}x^2-9x^4+12\end{align*}
5. \begin{align*}5x+2x^2-3x\end{align*}
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
TermDefinition
Polynomial A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Polynomial Function A polynomial function is a function defined by an expression with at least one algebraic term. | crawl-data/CC-MAIN-2017-17/segments/1492917125654.80/warc/CC-MAIN-20170423031205-00623-ip-10-145-167-34.ec2.internal.warc.gz | null |
What is Astaxanthin and why do you need it?
Unlike chlorophyll and Beta-Carotene, which are both found in terrestrial plants, Astaxanthin is found predominantly in marine life. A form of microalgae, Astaxanthin is consumed by many different types of aquatic life, and its intense red pigmentation results in these animals having red or pink flesh or outer shells.
Because of this, one of the highest concentrations of Astaxanthin, other than the Haematoccus Pluvialis algae (the truest form of the microalgae which fish eat) is found in Wild Pacific Sockeye Salmon.
This pigment is what gives the fish's flesh its signature deep red hue, and incredible antioxidant properties.
Scientists theorize that the antioxidant properties of natural Astaxanthin in salmon helps provide the endurance these remarkable animals need to swim upstream to spawn. Natural sources of marine life that contain Astaxanthin include Lobster, Shrimp, Crab, Trout, Algae & Krill. | <urn:uuid:e2a00416-03bf-4073-8d02-33c9c1593218> | {
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# Nails, Nails, Nails
## Snazzy Nails has the cheapest prices!!!
We simply stick to what we do best - creating beautiful nail designs and shaping those hands and feet. Come to Snazzy Nails the most luxurious nail salon and no longer will you have to be rich and famous to be treated like you are. Enjoy the luxury of your own personal spa-like while treating yourself to a magnificent manicure, and/or the perfect pedicure . We offer a unique nail care experience that relaxes you, relieves stress, improves well being and fits into your schedule and budget.
## Variables
Lets say x is number of hours.
Then y is the amount of money you will be paying/spending.
## Snazzy nails prices!!!
Equation of Line
Slope and y intercept form:
y=mx+b
y=7x+5
Standard form:
Ax + Bx + C = 0
0=7x-y+5
## Leading Nail Care Provider.
Equation of Line
Slope and y intercept form:
y=mx+b
y=5x+10
Standard form:
Ax + Bx + C = 0
0=5x-y+10
## Solution - Solving for a Point of Intersection
To find out when the two salons will charge the same number of products sold, we must solve the linear system, which is essentially finding a point of intersection.
1. We can solve by graphing
Using the equations given for each line, we can graph each line using the y intercept and rise over run for slope.
Snazzy Nails Salon
y = mx+b
y=7x+5
+5 is the y intercept so plot it
- Then use rise over run to graph the slope which is 7
- Graph the line
Leading Nail Care Provider
y = 5x + 10
- Repeat same steps
Then, using the 2 graphed lines, find the point of intersection. On the graph, the point of intersection is 2.5 hours spent on the nails which will result in you paying \$22.5.
2. You can also solve these linear systems by substitution (algebraically)
y = 5x + 10
y = 7x + 5
Sub in one equation into the other and solve for x
7x + 5 = 5x + 10
2x+5 = 10
2x = 5
x= 5/2
x = 2.5
Now sub x back into any equation and solve for y
y = 7(2.5) + 5
y = 22.8
Therefore, the point of intersection is (2.5 , 22.5)
## Snazzy Nail's Prices Compared to The Leading Nail Care Provider.
Legend
red line- leading nail care provider's prices
blue line- Snazzy nail's prices
## Conclusion
Snazzy nails is cheaper until 2.5 hours (where the break even point is).
The break even point is (POI) \$22.5 and 2.5 hours (2.5,22.5).
So at 2.5 hours both salons will charge you \$22.5.
After the break even point the leading nail care provider is cheaper. | crawl-data/CC-MAIN-2018-43/segments/1539583512161.28/warc/CC-MAIN-20181018235424-20181019020924-00149.warc.gz | null |
Rational Exponents
```Rational Exponents
The Rule for Rational Exponents
1
n
b b
n
1
3
64 64 4
3
How would we simplify this
expression?
What does the fraction exponent do to the
number?
9
1
2
The number can be written as a Radical
expression, with an index of the
denominator. 2
9
1
6
a
1
2
m
1
6
a a
1
2
6
m m
Exponents
5
b
w
Exponents
5
b b
1
5
ww
1
2
Negative exponents make inverses.
49
1
2
1
49
1
2
1
7
What if the numerator is not 1
Evaluate
2
5
32 32
5
2
For any nonzero real number b,
and integer m and n
Make sure the Radical express is real, no
b < 0 when n is even.
m
n
b b or
n
m
b
n
m
What if the numerator is not 1
Evaluate
2
5
32 32
5
5
2
5 2
2
2
5
10
What if the numerator is not 1
Evaluate
2
5
32 32
5
5
2
5 2
2 4
2
2
2
5
10
Simplify each expression
1
7
y y
x
2
3
4
7
Simplify
6
1
6
16 16
1
3
2
23
Simplify
1
6
6
16 16
1
3
2
23
1
4 6
(2 )
2
1
3
2
2
4
6
1
3
Simplify
1
6
6
16 16
1
3
2
23
1
4 6
(2 )
2
2
2
2
3
1
3
1
3
2
2
2
3
4
6
1
3
1
3
1
3
2 2 2 3 2
Simplify
6
4x
4
Simplify
6
1
6
4x4 4 x
4
6
Simplify
6
1
6
4x 4 x
4
x
2
1
2 6
4
6
4
6
Simplify
6
1
6
4x4 4 x
x
2
1
2 6
2
6
2 x
4
6
4
6
4
6
Simplify
6
1
6
4x4 4 x
2
1
2 6
2
6
4
6
x
4
6
4
6
1
3
2 x 2 x
2
3
Simplify
6
1
6
4x 4 x
4
2
1
2 6
2
6
x
4
6
4
6
4
6
1
3
2 x 2 x
2x
1
2 3
2
3
3 2x2
Practice Problems
``` | crawl-data/CC-MAIN-2022-05/segments/1642320304876.16/warc/CC-MAIN-20220125220353-20220126010353-00284.warc.gz | null |
Allergies are one of the fastest growing chronic diseases in childhood. The uptick may be due to lifestyle changes: As we spend more time indoors, exposure to allergens such as pets and dust mites increases. In the past 40 years there has been a dramatic increase in allergic conditions such as asthma, hayfever (allergic nasal symptoms), and food allergy, but not eczema, in the wealthier developed countries. The largest increase has occurred in the United States, United Kingdom, New Zealand and Australia. It is estimated that hayfever and asthma affect up to 40 percent of children, and some allergic manifestations may occur in as many as 65 percent of children in western populations. Children are selected for allergy studies because it avoids confusion with chronic infection, chronic obstructive lung disease (COPD), and symptoms due to cigarette smoke which are noted in adults. Allergic diseases are more common in urban vs. rural populations in the same countries and African Americans in the United States. Allergic food reactions have also increased significantly, but the true numbers are more difficult to determine because many children have food reactions which are not allergic.
It is commonly accepted that allergic diseases develop from environmental factors acting on genes in susceptible persons. One can have the gene to become allergic to cats, but if never exposed to cats, there will be no disease. It has been recognized for a long time that allergic diseases occur more commonly in the children of allergic parents. If both parents are allergic, children have a 50 percent risk of developing allergies. At the present time there is no single gene that has been identified that will predict if an individual will become allergic. Specific patterns, however, are associated with specific diseases. Eczema is primarily related to food allergic reactions. Allergic nasal symptoms are more commonly associated with allergic responses to pollen and outdoor molds.
Household Environmental Allergy Triggers
Asthma occurs more frequently in patients allergic to house dust mites, animals, cockroaches and the alternaria mold, a type of household mold. With house dust mites, the higher the exposure the more likely you are to develop the allergic antibody. Children with allergic parents require lower exposures than children from non-allergic parents to develop the allergic antibody. There does not appear to be a relationship between the amount of mite exposure and the development of asthma. With cat and dog exposures, the higher the exposure in the first year of life, the less likely are children to develop the allergic antibody or allergic diseases. These observations all indicate that many genes and complex environmental exposures are involved in allergic diseases.
It is now recognized that on exposure to foreign materials there are two major immune responses, the allergic and the protective response. Current evidence would indicate that by stimulating the protective response one can prevent and decrease the allergic response. The stimulation of the protective response before the development of the allergic response does not require knowing to what the child may become allergic and has been observed in a number of studies. High dog or cat exposure under one year of age can prevent the development of asthma, hay-fever or food allergies and the production of allergic antibodies to any allergen. The more infections and exposure to germs that children have, the less likely they are to become allergic. In several studies, there is less asthma in children who attend day care at ages 6-11 months vs. later. Older children in a family are more likely to have asthma than younger children, especially in larger families. This appeared to be related to more frequent infections in the younger children being exposed to older siblings with respiratory illnesses. In Italy and the Unites States, individuals with childhood hepatitis, toxoplasmosis or a specific stomach infection have fewer positive skin tests and allergic diseases. The same protection is also felt to explain the lower number of skin tests, asthma and allergic nasal symptoms in children growing up on farms. In these “dirty” environments there are more bacteria, bacteria products and parts, all of which are capable of stimulating the protective response similar to what is seen with infection and allergy vaccines.
Hygiene Hypothesis of Allergies
These observations have proposed the Hygiene Hypothesis as one explanation for the increase in allergies. It proposes that the emphasis on cleanliness has prevented the maturing of the immune response from an allergic to a protective response when faced with foreign substances. Pollen, mites, foods and molds are foreign but not harmful, and in the absence of a protective response the body responds with the allergic response. The Hygiene Hypothesis has also stimulated research to develop allergy vaccines which may be used in children of allergic parents to prevent allergic diseases and create more effective vaccines for children who show early manifestations of allergic symptoms. It has also decreased the emphasis on avoidance of foods and inhalants to prevent the development of allergic diseases. Once allergic however, avoidance is the best treatment.
Outside Environmental Factors
Another environmental factor contributing to the development of allergies is diesel fuel particles. Studies in Europe and the South Bronx have shown higher rates of allergic skin tests and asthma the closer you live to a major road and the Hunts Point Market. The Market is the largest in the world with diesel trucks being the major transportation of food products. Animal models demonstrate that diesel particles enhance the allergic response to pollen and outdoor molds.
Genetic Similarities of Obesity and Asthma
There are two epidemics occurring in developed countries. These are obesity and asthma, which are occurring in the same children and several mechanisms are postulated. One is that both share common genes. Another is that obese children are more likely to produce allergic responses then non-obese children. The third is that children with obesity and asthma have lower levels of Vitamin D3. This vitamin is also an antioxidant and may control the type of inflammation that occurs in allergic inflammation. Studies are being performed with Vitamin D3 in obese children with asthma and difficult to treat adult asthmatics. A simple treatment considered was exposure to sun to raise blood levels of Vitamin D3; this however was rejected by review boards because of the concern over skin cancer. We are fortunate to have good treatments for allergic diseases so mortality has decreased, but the cost to society and reduction in quality of life in individuals remains high. It is hoped as with other epidemics that improved allergy vaccines will be developed to prevent both disease and progression of disease.
Dr. John Condemi, Clinician, Allergy Asthma Immunology of Rochester | <urn:uuid:16514731-d3f4-47a7-91e0-dbb64d2041c9> | {
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List of Contents
What is the News?
According to a study, an increase in the number of rainy days leads to a downfall in economic output.
What is the study about?
The study was conducted to look at how rainfall patterns hurt the economy. The group compared daily rainfall data with subnational economic output from 77 countries between 1979 and 2019.
What are the key findings of the study?
An increase in the number of days with rainfall exceeding one millimeter led to a substantial decline in growth rates. A rise in extreme rainfall days contributed further to this loss.
Impact of Extreme Rainfall: The study suggests that increasing wet days and extreme rainfall will likely hit prosperous countries harder. This is because these countries rely more on the manufacturing and services sectors.
How will India be impacted if extreme rainfall happens?
In India, the agriculture sector is the most impacted. This is because of the quantum of the people involved and the economic share that agriculture provides.
However, this can be overcome by altering sowing dates, investing in irrigation and availing insurance in addition to changing crop varieties.
Moreover, the study also offers important lessons for India as the country aims to become a manufacturing hub. The country’s manufacturing sector is heavily dependent on supply chains, and supply chains are heavily disrupted during extreme weather events.
Note: Manufacturing Sector currently contributes roughly 17% to GDP. Its share in employment was 7.3% in 2020-2021.
Source: This post is based on the article ‘Rainfall changes could impact global manufacturing, services sectors’ published in Down To Earth on 22nd January 2022. | <urn:uuid:11f3ee98-5e5a-4c3b-a6e0-2b63b155eaa6> | {
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# Problem 83 Solution
If you confess there is probability p of two confessions and (1-p) probability that only you implicate your partner. If you don't confess there is probability p that neither of you implicate each other and a (1-p) probability that only you are implicated. Let s donate one unit of suffering. One year in jail yourself would cause you 2 units of suffering, and one year of your partner serving would cause you 1 unit of suffering.
The table below shows the total suffering given the four possibilities:
```
-----------------------
confess not confess
Y confess: 10*2+10=30 0+20=20
o
u not confess: 2*20+0=40 2*3+3=9
```
So given the probability p of your partner confessing if you confess the expected, or average, amount of suffering would be 30p + 20(1-p)=10p+20. If you don't confess the expected amount of suffering would be 40p + 9(1-p)=31p+9. To find the indifference point equate the two expressions:
10p+20 = 31p+9
11 = 21p
p=11/21 =~ 52.4%.
Thus if you think your partner's probability of confessing is greater than 11/21 you should confess, otherwise you should keep your mouth shut.
What is a much more interesting question, with no answer that fits everybody, is what would you do if you assumed your partner had the same morals and priorities as you, and that he assumes the same thing of you.
Michael Shackleford, A.S.A. | crawl-data/CC-MAIN-2019-13/segments/1552912202689.76/warc/CC-MAIN-20190322200215-20190322222215-00086.warc.gz | null |
A flu pandemic, also called an influenza pandemic, is an epidemic caused by an influenza virus, which covers a major portion of the world and infects a large proportion of humans. Though a flu pandemic occurs only infrequently, when one does occur it can cause many deaths. For instance, the 1918–1919 Spanish flu pandemic caused 20 to 50 million people to die.
The Spanish flu outbreak was not especially lethal in terms of number of deaths per total cases (by 1918 standards; however, the virus infected at least an estimated 500 million people). It was, however, very lethal to otherwise healthy adults ages twenty to forty-four years, as opposed to most flu outbreaks, which kill only the very young, the elderly, and people with weakened immune systems. Scientists and public health officials continue to study Spanish flu in the hopes of preventing a similar outbreak.
The Spanish flu virus caused one of the worst infectious disease pandemics ever recorded in modern history. Although the threats of some diseases, such as smallpox, have been contained by vaccination programs, influenza remains a difficult disease. There are worldwide outbreaks of influenza every year (approximately 300,000–500,000 people die of influenza or influenza complications each year—about 36,000 in the United States alone), and the flu typically reaches pandemic proportions (lethally afflicting an unusually high portion of the population) every ten to forty years. Prior to the declaration of a global pandemic of 2009 A H1N1 influenza in June 2009, the last influenza pandemic was the Hong Kong flu of 1968–1969, which caused an estimated one million deaths worldwide and killed approximately 33,000 Americans.
The influenza virus is highly mutable, so each year's flu outbreak presents the human body with a slightly different virus. Because of this, people do not build immunity to influenza. Vaccines are successful in protecting people against influenza, but vaccine manufacturers must prepare a new batch each year, based on their best supposition (hypothesis) of which particular virus will spread.
Most influenza viruses originate in Asia. Therefore, doctors, scientists, and public health officials closely monitor flu cases there to make the appropriate vaccine. The two main organizations tracking influenza are the U.S. Centers for Disease Control and Prevention (CDC) and the World Health Organization (WHO). The CDC and other government agencies had been preparing for a flu pandemic on the level of Spanish flu since the early 1990s.
Various symptoms are present with influenza. They include body aches, chills, cough, fatigue, fever, headache, runny or stuffy nose, and sore throat. A medical professional can prescribe antiviral drugs to treat these symptoms. Scientists are researching the causes of the various flu pandemics that have occurred on Earth. Two highly researched pandemics are the Spanish influenza and the 2009 A H1N1 influenza.
Social conditions at the time probably contributed to the remarkable power of the disease. The flu struck just at the end of World War I, when thousands of soldiers were moving from America to Europe and across that continent. In a peaceful time, sick people may have gone home to bed, and thus passed the disease only to their immediate family. However, in 1918, men with the virus were packed in already crowded hospitals and troop ships. The unrest and devastation left by the war probably hastened the spread of Spanish flu. So it is possible that if a similarly virulent virus were to arise again soon, especially with modern anti-viral medicines, it would not be as deadly.
Researchers are concerned about a return of Spanish flu because little is known about what made it so virulent. The flu virus was not isolated until 1933, and since then there have been several efforts to collect and study the 1918 virus by exhuming corpses in Alaska and Norway, where bodies were preserved in permanently frozen ground. In 1997, a Canadian researcher, Kirsty Duncan (1966–), was able to extract tissue samples from the corpses of seven miners who had died of Spanish flu in October 1918 and were buried in frozen ground on a tiny island off Norway. Duncan's work allowed scientists at several laboratories around the world to do genetic work on the Spanish flu virus. It was subsequently determined by researchers that the Spanish flu was a mixture of genes, including avian genes.
The 2009 pandemic H1N1 virus proved to be less lethal than many other flu viruses (for example it is not as lethal as the H5N1, Spanish, or Hong Kong viruses), and infectious disease experts predicted that the 2009 influenza pandemic would not approach the severity of prior pandemics such as the 1918–1919 Spanish flu. Priorities for responding to the pandemic included manufacturing and delivering sufficient quantities of antiviral drugs and vaccine specific to the 2009 H1N1 virus.
The influenza virus is believed to originate in migratory waterfowl, particularly ducks. Ducks carry influenza viruses without becoming ill. They excrete the virus in their feces. When their feces collect in water, other animals can become infected. Domestic turkeys and chickens can easily become infected with influenza virus borne by wild ducks. However, most avian (bird-borne) influenza does not pass to humans, or if it does, is not particularly virulent. Other mammals, too, can pick up influenza from either wild birds or domestic fowl. Whales, seals, ferrets, horses, and pigs are all susceptible to bird-borne viruses. When the virus moves between species, it may mutate. Human influenza viruses may sometimes pass from ducks to pigs to humans.
A flu outbreak among chickens in Hong Kong in 1997 eventually killed six people, but the epidemic was stopped by the quick slaughter of millions of chickens in the area. This virus identified was classified as an avian flu (bird flu), H5N1 strain of influenza. As of August 2010, H5N1 influenza had claimed approximately 300 human lives, and the lethal virus had spread geographically to Europe and Africa. Up to that point, only a few cases of human-to-human transmission of H5N1 had been documented. However, because of its lethality, the H5N1 virus was closely monitored by epidemiologists for signs that the virus could mutate in such a way that it could transmit easily between humans, a necessary step toward pandemic level outbreaks.
Antiviral drugs are used to treat influenza. They are prescription drugs that fight the flu virus found within the human body. Antiviral drugs affect viral infections, which are different from antibiotics, which fight bacterial infections. Both drugs, however, lessen the associated symptoms and shorten the time that one is sick. Antiviral drugs also prevent serious flu complications, like pneumonia. In addition, for persons who are exceptionally susceptible for having serious complications from the flu, antiviral drugs minimize these people's risk from a more serious illness or even death.
More detailed information is provided on the flu.gov website under Pandemic Flu ( http://www.flu.gov/planning-preparedness/federal/index.html ).
People can minimize their risk for getting influenza during a pandemic by following several simple guidelines. When these guidelines are followed the prognosis for surviving such an pandemic is good. These guidelines are:
Enacting controls on pig and poultry farms may be an important way to prevent the rise of a new influenza pandemic. Some influenza researchers recommend that pigs and domestic ducks and chickens not be raised together. Separating pigs and fowl at live markets may also be a sensible precaution. With the concentration of poultry and pigs at factory farms, it is important for farmers, veterinarians, and public health officials to monitor these farms for influenza. Any action to control flu must be an international effort, because the influenza virus moves rapidly without respect to national borders.
While not completely able to prevent influenza, flu vaccines are able to minimize one's chances of getting the flu. Medical professionals consider flu vaccines as the better way to prevent influenza when compared with antiviral drugs, which are normally used to fight the illness once it occurs.
See also Avian flu ; Centers for Disease Control and Prevention ; H1N1 influenza A (2009) ; Pandemic ; U.S. Department of Health and Human Services ; World Health Organization .
Beck, Eduard J. The HIV Pandemic: Local and Global Implications. New York: Oxford University Press, 2008.
Bristow, Nancy K. American Pandemic: The Lost Worlds of the 1918 Influenza Epidemic. Oxford: Oxford University Press, 2012.
Dehner, George. Influenza: A Century of Science and Public Health Response. Pittsburgh: University of Pittsburgh Press, 2012.
Duncan, Kirsty. Hunting the 1918 Flue: One Scientist's Search for a Killer Virus. Toronto: University of Toronto Press, 2006.
Goldsmith, Connie. Influenza: The Next Pandemic. Minneapolis, MN: Twenty-first Century Books, 2007.
Hays, J. N. Epidemics and Pandemics: Their Impacts on Human History. Santa Barbara, CA: ABC-CLIO, 2005.
Langwith, Jacqueline. Pandemics. Farmington Hills, MI: Greenhaven Press, 2012.
Ryan, Jeffrey R. Pandemic Influenza: Emergency Planning and Community Preparedness. Boca Raton, FL: CRC Press, 2009.
Sipress, Alan. The Fatal Strain: On the Trail of the Coming Avian Flu Pandemic. New York: Viking, 2009.
Influenza at the Human-Animal Interface (HAI). World Health Organization. http://www.who.int/influenza/human_animal_interface/en/index.html (accessed September 12, 2012).
Epidemic and Pandemic Alert and Response. Flu.gov, U.S. Department of Health & Human Services. http://www.flu.gov/ (accessed September 12, 2012).
Global Alert and Response (GAR). World Health Organization. http://www.who.int/csr/en/ (accessed September 12, 2012).
Pandemic Flu Preparedness Tools. Centers for Disease Control and Prevention. (June 21, 2012). http://www.cdc.gov/flu/antivirals/whatyoushould.htm (accessed September 12, 2012).
Preparing for a Pandemic. Scientific American. (October 24, 2005). http://www.scientificamerican.com/article.cfm?id=preparing-for-a-pandemic-2005-11 (accessed September 12, 2012).
American Medical Association, 515 N. State St., Chicago, IL, 60654, (800) 621-8335, http://www.ama-assn.org/ .
Centers for Disease Control and Prevention, 1600 Clifton Rd., Atlanta, GA, 30333, (800) 232-4636, [email protected], http://www.cdc.gov/ .
World Health Organization, Avenue Appia 20, Geneva, Switzerland, 1211 27, 41 22 791-2111, Fax: 41 22 7913111, [email protected], http://www.who.int/en/ .
Revised by William A. Atkins, BB, BS, MBA | <urn:uuid:b214e9f3-e348-44a0-bef5-33c32ef06e41> | {
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WW1 led to many changes in the social fabric all countries involved. In Canada, the boys were shipped overseas to fight, leaving the women to take over the various chores they left behind.
Before the war, women had few rights - they could not vote or hold political office, only a few worked outside the house,
Only 14 years before the War, the Marriage Property Act made ti possible for a woman to control her own property and wages separately from her spouse, while being jointly responsible for child support. Teaching was the only profession which provided a pension. See this reference for milestones for women in Canada.
One of the most important supportive tasks which needed to be maintained was in agriculture. While women did train to become nurses and went overseas, others remained at home and assumed the farming duties needed t maintain the civilian population at home and the fighting men overseas. June Hitchcox describes her experience as a member of the Farm Service Force.in Ontario. | <urn:uuid:7cf86cc0-9cfb-4491-8211-abf41c042194> | {
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# What Is a List of Some Benchmark Fractions?
By Staff WriterLast Updated Mar 28, 2020 5:21:11 AM ET
A list of benchmark fractions include 1/4, 1/3, 1/2, 2/3 and 3/4. Benchmark fractions are common fractions that are used for comparison to other numbers. For example, the benchmark fraction 1/10 is often used because of how it relates to decimals.
Benchmark fractions are useful to know because of how they relate to common percentages and decimals. Every fraction can be converted into a decimal by dividing the denominator, or the bottom number, into the numerator, the top number. The decimal can then be converted into a percentage by multiplying by 100. A list of some benchmark fractions and their equivalents are:
• 1/2 = 50 percent = 0.5
• 1/4 = 25 percent = 0.25
• 2/5 = 40 percent = 0.4
• 3/8 = 37.5 percent = 0.375
• 7/10 = 70 percent = 0.7
• 1/9 = 11.11 percent = 0.111
• 1/8 = 12.5 percent = 0.125
• 4/5 = 80 percent = 0.8
More From Reference | crawl-data/CC-MAIN-2021-25/segments/1623487648373.45/warc/CC-MAIN-20210619142022-20210619172022-00310.warc.gz | null |
# What is the equation of the normal line of f(x)=2x^3-8x^2+2x-1 at x=-1?
Feb 9, 2016
$y + 13 = - \frac{1}{24} \left(x + 1\right)$
#### Explanation:
First, find the point the normal line will intercept by finding the function value at $x = - 1$.
f(-1)=2(-1)^3-8(-1)^2+2(-1)-1=-2-8-2-1=ul(-13
The normal line will pass through the point $\left(- 1 , - 13\right)$.
Before we can find the slope of the normal line, we must first find the slope of the tangent line. The slope of the tangent line is equal to the value of the function's derivative at $x = - 1$.
Find the derivative of the function through the power rule:
$f \left(x\right) = 2 {x}^{3} - 8 {x}^{2} + 2 x - 1$
$f ' \left(x\right) = 6 {x}^{2} - 16 x + 2$
The slope of the tangent line is
f'(-1)=6(-1)^2-16(-1)+2=6+16+2=ul(24
Since the tangent line and normal line are perpendicular, their slopes will be opposite reciprocals. The opposite reciprocal of $24$ is $- \frac{1}{24}$.
We can relate the information we know about the normal line as a linear equation in point-slope form, which takes a point $\left({x}_{1} , {y}_{1}\right)$ and slope $m$:
$y - {y}_{1} = m \left(x - {x}_{1}\right)$
Since the normal line passes through $\left(- 1 , - 13\right)$ and has slope $- \frac{1}{24}$, its equation is
$y + 13 = - \frac{1}{24} \left(x + 1\right)$
Graphed are the function and its normal line:
graph{(2x^3-8x^2+2x-1-y)(y+13+(x+1)/24)=0 [-5, 7, -18.16, 2.12]} | crawl-data/CC-MAIN-2024-38/segments/1725700651714.51/warc/CC-MAIN-20240916212424-20240917002424-00392.warc.gz | null |
Cortical visual impairment (CVI) is a term used to describe visual impairment that occurs due to brain injury. CVI differs from other types of visual impairment which are due to physical problems with the eyes. CVI is caused by damage to the visual centers of the brain, which interferes with communication between the brain and the eyes. The eyes are able to see, but the brain is not interpreting what is being seen.
Cortical visual impairment (CVI) is often referred to by other terms including: cerebral visual impairment, neurological visual impairment, brain damage related visual impairment and so forth. All of these terms refer to visual dysfunction resulting from injury to visual centers of the brain. We will always refer to it as cortical visual impairment or CVI.
Typical characteristics of CVI:
- Preference for a specific color. You may have noticed that your child seems to prefer looking at a certain color. Bright red and yellow are often favorite colors, but some children prefer other bright colors such as blue, green, or pink.
- Need or preference for movement. Many children with CVI require movement in order to see an object. For example, it may be easier for them to look at a pinwheel or a swaying balloon.
- Delayed response when looking at objects (visual latency). It may take time for a child with CVI to look at an object. Often they will look at an object and then look away. For this reason it is important to give your child enough time when presenting an object.
- Difficulty with visual complexity. Children with CVI need simplicity. First, it is important that the object presented is simple. For example, a single colored stuffed animal, like Elmo, is preferable to one with multiple colors. Likewise, it is important that the background and the environment are simple. For example, putting a solid black cloth behind a single colored toy helps to reduce visual clutter. Creating a simple environment is a matter of eliminating noise and anything else that might distract from the visual task.
- Light-gazing and nonpurposeful gazing. Often, children with CVI will stare at light. They may be seen gazing out the window or up at a ceiling light. They may also appear as if looking at things that are not there, or looking at things without intent.
- Visual field preferences. Most children with CVI will prefer to look at objects in a particular direction. For example, they may see an object better when it is presented in their periphery, or may turn their head to see an object.
- Distance vision impaired. Some children with CVI have trouble seeing far away. This is related to the preference for visual simplicity. Objects far away may be lost in visual clutter.
- Visual blink reflex is absent or impaired. When an object comes too close to the eyes, or touches the bridge of the nose, many CVI children have an absent or delayed protective blink response.
- Preference for familiar objects. Because it is difficult for CVI children to process the information that the eyes see, they often prefer familiar objects that the brain easily recognizes and has processed before.
- Impaired visually guided reach. The ability to look at an object while reaching for it is impaired. Often CVI children will look away from the object and then reach for it.
The three phases of CVI:
Dr. Roman-Lantzy, author of Cortical Visual Impairment: An Approach to Assessment and Intervention, divides CVI into three phases. Most children start in Phase I, which means that most of the CVI characteristics are present. As a child progresses through the three phases many of the characteristics begin to resolve. This process can take several years and requires diligence and persistence. Children in Phase III approach near normal vision to varying degrees and this may even result in literacy.
CVI often goes undiagnosed. It may go undiagnosed by an ophthalmologist because the structure of the eye is often normal. Many parents are told that there is no way to know what, or how much, their child can see. They are often told that there is nothing that can be done, and to just wait and see. In our experience, very few medical professionals are aware of CVI. Frequently, parents are the first to notice some visual responses in their children. It is our hope that with this website we can empower those parents to help their children learn to see.
There is hope!
The wonderful reality of CVI is that it can, and usually does, get better with appropriate intervention. A study conducted by Dr. Roman-Lantzy found that, in a select group of children with CVI who had highly motivated parents, 97% went from Phase I to Phase III in an average of 3.7 years. Some vision specialists or teachers of the visually impaired (TVI) are knowledgeable about CVI and can help with assessment and intervention strategies. Even without the assistance of vision specialists there is a lot that you as a parent can do.
This page is now printable. This allows parents to print the basics about CVI for any doctors, therapists, caregivers, or family members who may not know about CVI. Our website address is included at the bottom of the page, so anyone who receives it can visit us to learn more, if they’d like. What is CVI – printable PDF file. | <urn:uuid:e6dc57b1-24c7-4c32-8248-11089703cf45> | {
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# Normal distribution
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### Normal distribution
1. 1. Most popular continuous probability distribution is normal distribution. It has mean μ & standard deviation σ Deviation from mean x- μ Z = --------------------------- = -------------- Standard deviation σ Graphical representation of is called normal curve
2. 2. Standard deviation( σ ) Standard deviation is a measure of spread ( variability ) of around Because sum of deviations from mean is always zero , we measure the spread by means of standard deviation which is defined as square root of ∑ (x- μ) 2 ∑ (x- xbar) 2 Variance (σ2) = -------------= ---------------- N n-1 σ2 = variance
3. 3. Interpretation of sigma (σ ) 1.Sigma (σ ) – standard deviation is a measure of variation of population 2.Sigma (σ ) – is a statistical measure of the process’s capability to meet customer’s requirements 3.Six sigma ( 6σ ) – as a management philosophy 4.View process measures from a customer’s point of view 5.Continual improvement 6.Integration of quality and daily work 7.Completely satisfying customer’s needs profitably
4. 4. Use of standard deviation( σ ) Standard deviation enables us to determine , with a great deal of accuracy , where the values of frequency distribution are located in relation to mean. 1.About 68 % of the values in the population will fall within +- 1 standard deviation from the mean 2.About 95 % of the values in the population will fall within +- 2 standard deviation from the mean 3.About 99 % of the values in the population will fall within +- 3 standard deviation from the mean
5. 5. z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
6. 6. 1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
7. 7. 2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993 3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995 3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997 3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998
8. 8. SIGMA Mean Centered Process Mean shifted ( 1.5) Defects/ million % Defects/ million % 1 σ 317400 31.74 697000 69.0 2 σ 45600 4.56 308537 30.8 3 σ 2700 .26 66807 6.68 4 σ 63 0.0063 6210 0.621 5 σ .57 0.0000 6 233 0.0233 6 σ .002 3.4 0.0003 4
9. 9. Thus , if t is any statistic , then by central limit theorem variable value – Average Z = -------------- Standard deviation or standard error x- μ Z = -------------- σ
10. 10. Properties 1. Perfectly symmetrical to y axis 2. Bell shaped curve 3. Two halves on left & right are same. Skewness is zero 4. Total area1. area on left & right is 0.5 5. Mean = mode = median , unimodal 6. Has asymptotic base i.e. two tails of the curve extend indefinitely & never touch x – axis ( horizontal )
11. 11. Importance of Normal Distribution 1. When number of trials increase , probability distribution tends to normal distribution .hence , majority of problems and studies can be analysed through normal distribution 2. Used in statistical quality control for setting quality standards and to define control limits
12. 12. Hypothesis : a statement about the population parameter Statistical hypothesis is some assumption or statement which may or may not be true , about a population or a probability distribution characteristics about the given population , which we want to test on the basis of the evidence from a random sample
13. 13. Testing of Hypothesis : is a procedure that helps us to ascertain the likelihood of hypothecated population parameter being correct by making use of sample statistic A statistic is computed from a sample drawn from the parent population and on the basis of this statistic , it is observed whether the sample so drawn has come from the population with certain specified characteristic
14. 14. Procedure / steps for Testing a hypothesis 1. Setting up hypothesis 2. Computation of test statistic 3. Level of significance 4. Critical region or rejection region 5. Two tailed test or one tailed test 6. Critical value 7. Decision
15. 15. Hypothesis : two types 1. Null Hypothesis H0 2. Alternative Hypothesis H1 Null Hypothesis asserts that there is no difference between sample statistic and population parameter & whatever difference is there it is attributable to sampling errors Alternative Hypothesis : set in such a way that rejection of null hypothesis implies the acceptance of alternative hypothesis
16. 16. Null Hypothesis Say , if we want to find the population mean has a specified value μ0 H0 : μ = μ0 Alternative Hypothesis could be i. H1 : μ ≠ μ0 ( i.e. μ > μ0 or μ < μ0 ) ii. H1 : μ > μ0 iii. H1 : μ < μ0 iv. R. A. Fisher “Null Hypothesis is the hypothesis which is to be tested for possible rejection under the assumption that it is true
17. 17. 4. level of significance :is the maximum probability ( α ) of making a Type I error i.e. : P [ Rejecting H0 when H0 is true ] Probability of making correct decision is ( 1 - α ) Common level of significance 5 % ( .05 ) or 1 % ( .01 ) For 5 % level of significance ( α = .05 ) , probability of making a Type I error is 5 % or .05 i.e. : P [ Rejecting H0 when H0 is true ] = .05 Or we are ( 1 - α or 1-0.05 = 95 % ) confidence that a correct decision is made When no level of significance is given we take α = 0.05
18. 18. 5.Critical region or rejection region :the value of test statistic computed to test the null hypothesis H0is known as critical value . It separates rejection region from the acceptance region
19. 19. 6.Two tailed test or one tailed test : Rejection region may be represented by a portion of the area on each of the two sides or by only one side of the normal curve , accordingly the test is known as two tailed test ( or two sided test ) or one tailed ( or one sided test )
20. 20. Two tailed test :where alternative hypothesis is two sided or two tailed e.g. Null Hypothesis H0 : μ = μ0 Alternative Hypothesis H1 : μ ≠ μ0 ( i.e. μ > μ0 or μ < μ0 )
21. 21. One tailed test :where alternative hypothesis is one sided or one tailed two types a. Right tailed test :- rejection region or critical region lies entirely on right tail of normal curve b. Left tailed test :- rejection region or critical region lies entirely on left tail of normal curve
22. 22. Right tailed : Null Hypothesis H0 : μ = μ0 Alternative Hypothesis H1 : μ > μ0 Left tailed : Null Hypothesis H0 : μ = μ0 Alternative Hypothesis H1 : μ < μ0 Right tailed
23. 23. 7.Critical value : value of sample statistic that defines regions of acceptance and rejection Critical value of z for a single tailed ( left or right ) at a level of significance α is the same as critical value of z for two tailed test at a level of significance 2α .
24. 24. Critical value (Zα ) Level of significance 1 % 5 % 10 % Two tailed test [Zα ] = 2.58 [Zα ] = 1.96 [Zα ] = 1.645 Right tailed test Zα = 2.33 Zα = 1.645 Zα= 1.28 Left tailed test Zα = - 2.33 Zα = -1.645 Zα = - 1.28
25. 25. S.No. Confidence level (1- α ) Value of confidence coefficient Z α ( two tailed test) 1 90 % 1.64 2 95 % 1.96 3 98 % 2.33 4 99 % 2.58 5 Without any reference to confidence level 3.00 6 α is level of significance which separates acceptance & rejection level
26. 26. 8. Decision : 1. if mod .Z < Zα accept Null Hypothesis test statistic falls in the region of acceptance 2. if mod .Z > Zα reject Null Hypothesis
27. 27. Q1.Given a normal distribution with mean 60 & standard deviation 10 , find the probability that x lies between 40 & 74 Given μ= 60 , σ =10 P ( 40 < x < 74 ) = P ( -2 < z< 1.4 ) = P ( -2< z < 0 ) + P ( 0 < z < 1.4 ) = 0.4772+ 0.4192 = 0. 8964
28. 28. Q2.In a project estimated time of completion is 35 weeks. Standard deviation of 3 activities in critical paths are 4 , 4 & 2 respectively . calculate the probability of completing the project in a. 30 weeks , b. 40 weeks and c. 42 weeks
29. 29. Test of significance Mean Null –there is no significance difference between sample mean & population mean or The sample has been drawn from the parent population Deviation from mean xbar- μ Z = --------------------------- = -------------- Standard Error Standard Error xbar = sample mean μ = population mean
30. 30. 1. Standard Error of mean = σ / √ n When population standard deviation is known σ = standard deviation of the population n = sample size 2. Standard Error of mean = s / √ n When sample standard deviation is known s = standard deviation of the sample n = sample size
31. 31. Proprtion Null –there is no significance difference between sample proportion & population proportion or The sample has been drawn from a population with population proportion P Null hypothesis H0 : P = P0 where P0 is particular value of P Alternate hypothesis H1 : P ≠ P0 ( i.e. P > P0 or P < P0 )
32. 32. P*(1-P) Standard error of proportion (S.E.(p)) = √ ------------ n Deviation from proprtion p-P Test statistic Z = --------------------------- = -------- Standard Error (p) S.E.(p)
33. 33. Q3. a sample of size 400 was drawn and sample mean was 99. test whether this sample could have come from a normal population with mean 100 & standard deviation 8 at 5 % level of significance
34. 34. Ans. Given xbar = 99 , n = 400 μ = 100 , σ = 8 1.Null hypothesis sample has come from a normal population with mean = 100 & s.d. = 8 Null hypothesis H0 : μ = 100 Alternate hypothesis H1 : μ ≠ μ0 ( i.e. μ > μ0 or μ < μ0 ) Two tail test so out of 5% , 2.5 % on each side ( left hand & right hand) 2.calculation of Test statistic Standard error (s.e.) of xbar = σ / √ n = 8/√ 400 = 8/20= 2/5 xbar- μ 99-100 Test statistic Z = ----------- = -------- = -5/2 =- 2.5 S.E. 2/5
35. 35. Mod z = 2.5 3. level of significance 5 % i.e.value of α = .05 ( hence , level of confidence = 1- α = 1-0.05 = 0.95 or 95%) 4. Critical value (since it is Two tail test so out of 5% , we take two tails on each side i.e.2.5 % on each side = 0.025 ( left hand & right hand) = read from z –table value corresponding to area = 0.5 -0.025 = 0.4750 ( 0.4750 on both sides i.e. 2* 0.4750 = 0.95 area which means 95% confidence) = value of z corresponding to area is 1.96
36. 36. 5. Decision – since mod value of z is more than critical value Null Hypothesis is rejected & alternate hypothesis is accepted Sample has not been drawn from a normal population with mean 100 &s.d. 8
37. 37. Q4. the mean life time of a sample of 400 fluorescent light tube produced by a company is found to be 1570 hours with a standard deviation of 150 hrs. test the hypothesis that the mean life time of the bulbs produced by the company is 1600 hrs against the alternative hypothesis that it is greater than 1600 hrs at 1 &% level of significance
38. 38. Ans. Given xbar = 1570 , n = 400 μ = 1600 , standard deviation of sample mean s= 150 = 8 Null hypothesis : mean life time of bulbs is 1600 hrs i.e. Null hypothesis H0 : μ = 1600 Alternate hypothesis H1 : μ > 1600 i.e. it is a case of right tailed test 2.calculation of Test statistic Standard error (s.e.) of xbar = s / √ n 150/√ 400 = 150/20= 7.5
39. 39. xbar- m 1570-1600 Z = ----------- = -------------- = -30/7.5= - 4 S.E. 7.5 Mod z = 4 3. level of significance 1 % i.e. value of α = .01 ( hence , level of confidence = 1- α = 1-0.01 = 0.99 or 99%)
40. 40. 4. Critical value (since it is right tail test so out of 1% , we take 1 % only one side = 0.01 on right hand) = read from z –table value corresponding to area = 0.5 -0.01 = 0.4900 ( 0.49 on one side together with +0.5 makes total 0.99 which means 99% confidence) = value of z ( corresponding to area 0 .49 is 2.33 5. Decision – since mod value of z is more than critical value Null Hypothesis is rejected & alternate hypothesis is accepted Hence mean life time of bulbs is greater than 1600 hrs
41. 41. Q5.in a sample of 400 burners there were 12 whose internal diameters were not within tolerance . Is this sufficient to conclude that manufacturing process is turning out more than 2 % defective burners. Take α = .05
42. 42. Given P= 0.002 Q= 1-P =1-0.02 =0. 98 & p= 12/400 = 0.03 Null hypothesis H0 : P = process is under control P ≤ 0.02 Alternate hypothesis H1 : P > 0.02 Left tail test Calculation of Standard error of proportion P*(1-P) 0.02*0.98 (S.E.(p)) = √ ------------ = √-------------- n 400
43. 43. Deviation 0.03-0.02 0.001 Z = ----------- = ------------ = -------- = 1.429 S.E.(p) √ (0.02*0.98) / 400 0.007 3. level of significance 5 % i.e.value of α = .05 ( hence , level of confidence = 1- α = 1-0.01 = 0.95 or 95%)
44. 44. 4. Critical value (since it is left tail test so out of 5% , we take full 5 % only one side = 0.05 on left hand) = read from z –table value corresponding to area = 0.5 -0.05 = 0.4500 ( 0.45 on one side together with +0.5 makes total 0.95 which means 95% confidence) = value of z ( corresponding to area 0 .45 is 1.645 5. Decision – since mod value of z is less than critical value Null Hypothesis is accepted Hence process is not out of control
45. 45. Q6. a manufacturer claimed that at least 95 % of the equipment which he supplied is conforming to specifications. A examination of sample of 200 pieces of equipment revealed that 18 were faulty. Test his claim at level of significance i.) 0.05 ii.) 0.01
46. 46. Given P= 0.95 Q= 1-P =1-0.95 =0. 05 n= 200 p= 18/200 = - (200-18) / 200 = 182/200 = 0.91 Null hypothesis H0 : P = process is under control P = 0.95 Alternate hypothesis H1 : P < 0.95 Left tail test
47. 47. P*(1-P) 0.95*0.05 S.E.(p) = √ ------------ = √--------- n 200 0.91-0.95 -0.04 Z = ------------ = --------- = -2.6 √ (0.02*0.98) / 200 0.0154
48. 48. 3a. level of significance 5 % i.e.value of α = .05 ( hence , level of confidence = 1- α = 1-0.05 = 0.95 or 95%) 4a. Critical value (since it is right tail test so out of 5% , full 5 % on one side = 0.05 on right hand) = read from z –table value corresponding to area = 0.5 -0.05 = 0.4500 ( 0.45 on one side together with +0.5 makes total 0.95 which means 95% confidence) = value of z ( corresponding to area 0 .45 is 1.645) 5a. Decision – since mod value of z is more than critical value Null Hypothesis is rejected Manufacturer’s claim is rejected at 5 % level of significance
49. 49. 3b. level of significance 1 % i.e.value of α = .01 ( hence , level of confidence = 1- α = 1-0.01 = 0.99 or 99%) 4b. Critical value (since it is right tail test so out of 1% , we take 1 % only one side = 0.01 on right hand) = read from z –table value corresponding to area = 0.5 -0.01 = 0.4900 ( 0.49 on one side together with +0.5 makes total 0.99 which means 99% confidence) = value of z ( corresponding to area 0 .49 is 2.33
50. 50. 5b. Decision – since mod value of z is more than critical value Null Hypothesis is rejected Manufacturer’s claim is rejected at 1 % level of significance
51. 51. Campus Overview 907/A Uvarshad, Gandhinagar Highway, Ahmedabad – 382422. Ahmedabad Kolkata Infinity Benchmark, 10th Floor, Plot G1, Block EP & GP, Sector V, Salt-Lake, Kolkata – 700091. Mumbai Goldline Business Centre Linkway Estate, Next to Chincholi Fire Brigade, Malad (West), Mumbai – 400 064.
52. 52. Thank You | crawl-data/CC-MAIN-2016-30/segments/1469257827080.38/warc/CC-MAIN-20160723071027-00100-ip-10-185-27-174.ec2.internal.warc.gz | null |
Recursion is one of the tough programming technique to master. Many programmers working on both Java and other programming language like C or C++ struggles to think recursively and figure out recursive pattern in problem statement, which makes it is one of the favorite topic of any programming interview. If you are new in Java or just started learning Java programming language and you are looking for some exercise to learn concept of recursion than this tutorial is for you. In this programming tutorial we will see couple of example of recursion in Java programs and some programming exercise which will help you to write recursive code in Java e.g. calculating Factorial, reversing String and printing Fibonacci series using recursion technique. For those who are not familiar with recursion programming technique here is the short introduction: "Recursion is a programming technique on which a method call itself to calculate result". Its not as simple as it look and mainly depends upon your ability to think recursively. One of the common trait of recursive problem is that they repeat itself, if you can break a big problem into small junk of repetitive steps then you are on your way to solve it using recursion.
How to solve problem using Recursion in Java
In order to solve a problem using recursion in Java or any other programming language e.g. C or C++, You must be able to figure out :
1) Base case, last point which can be resolve without calling recursive function e.g. in case of Fibonacci series its
1 and 2 where result will be 1. In case of recursive power function its zero power which is equal to 1 or in case of calculating Factorial its factorial of zero which is equal to 1.
2) With every recursive method call, Your problem must reduce and approach to base case. If this is not the case than you won't be able to calculate result and eventually die with java.lang.StackOverFlowError
Recursion Programming Example in Java
In our programming example of recursion in Java we will calculate Fibonacci number of give length using recursion. In case of Fibonacci number current number is sum of previous two number except first and second number which will form base
case for recursive solution.
If you look above example of recursion in Java you will find that we have a base case where program returns result before calling recursive function and than with every invocation number is decreased by 1. This is very important to reach solution using recursive technique.
Programming Exercise to solve using Recursion
Here are few more programming exercise to learn Recursion programming technique in Java programming language. This exercise are solely for practicing. In order to understand Recursion properly you must try to think recursive e.g. look tree as collection of small tree, look string as collecting of small String, look staircases as collection of small staircase etc. Any way try to solve following programming exercise by using Recursion programming technique for better understanding
1. Print Fibonacci series in Java for a given number, see here for solution
2. Calculate factorial of a give number in Java, see here for solution of this programming exercise
3. Calculate power of a give number in java
4. Reverse a String using recursion in Java, see here for solution
5. Find out if there is a loop in linked list using recursion
This was simple introduction of Recursion programming technique to Java programmer with most basic examples. There are lot more to learn on Recursion including different types of recursion e.g. tail recursion, improving performance of recursive algorithm using memoization or caching pre calculated result etc. Its also recommended not to use recursive method in production code instead write iterative code to avoid any stackoverflow error.
Other programming tutorials from Javarevisited Blog | <urn:uuid:4ab8b3a8-9fae-4800-a420-e5e8b236c6e1> | {
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Not only were the 1930's in America a time of great economic strain and change, but they were also a time of hostile cultural conflict. The nation's universities saw great intellectual sparring between philosophical rationalists and empirical, scientific naturalists. The rationalist school, filled with neo-Aristotilians like Mortimer J. Adler and Robert M. Hutchins, promoted a philosophical rationalism that claimed reason can discover immutable metaphysical principles that are the foundations for the rationally true and ethically good. The scientific naturalists, like Anton Carlson and John Dewey, claimed that there were no absolute principles in the universe and that man should govern his actions with an empirical, scientific problem-solving pragmatism.
This philosophical debate was not restricted to the Academy. The philosophy behind the New Deal was primarily a pragmatic one, and people who saw this approach as a departure from their accepted way of life felt anxiety at the change. What was at stake in this conflict was more than mere political power; the theory of education, the way of viewing the past, the way of respecting authority, and Americans' very theory of their own identity hinged in the balance of this intellectual and sociological debate.
This conflict between two opposing philosophical camps can be seen in two novels of the 1930's, Absalom, Absalom! by William Faulkner and Gone With the Wind by Margaret Mitchell. In both novels characters attempt to function in a "new" world formed in the aftermath of the Civil War. In many ways the South of Reconstruction resembled all of America during the Depression, and the choices that Faulkner's and Mitchell's characters are forced to make would have resonated especially with their 1930's readers. | <urn:uuid:ad549691-f1dd-4358-b348-18be3193cd8d> | {
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# Central Projection of the Sphere
Given the parameterization of the unit sphere $x^2+y^2+z^2=1$ as
$x = \displaystyle\frac{u}{\sqrt{1+u^2+v^2}}$
$y = \displaystyle\frac{v}{\sqrt{1+u^2+v^2}}$
$z = \displaystyle\frac{1}{\sqrt{1+u^2+v^2}}$
Find $ds^2=dx^2+dy^2+dz^2$ and using the metric computer the area of the hemisphere $z\geq0$
I got:
$dx = \displaystyle\frac{v^2+1}{(u^2+v^2+1)^{3/2}}du-\displaystyle\frac{uv}{(u^2+v^2+1)^{3/2}}$dv
$dy = -\displaystyle\frac{uv}{(u^2+v^2+1)^{3/2}}du+\displaystyle\frac{u^2+1}{(u^2+v^2+1)^{3/2}}dv$
$dz = \displaystyle\frac{-u}{(u^2+v^2+1)^{3/2}}du-\displaystyle\frac{v}{(u^2+v^2+1)^{3/2}}dv$
And
$dx^2+dy^2+dz^2= \displaystyle\frac{(v^4+2v^2+1+u^2+u^2v^2)du^2+(u^4+2u^2+1+v^2+u^2v^2)dv^2+(-2uv-2u^3v-2uv^3)dudv}{(u^2+v^2+1)^3}$
But I can't see an obvious simplification
-
Spherical coordinates are for spheres. I gather you may not have such freedom here. That said: notice that you can factor the $dudv$ term $-2uv(1+u^2+v^2)$. I'm not totally sure your multiplication is correct (I've not checked it all) – James S. Cook Oct 2 '12 at 1:51
yes i have to deal with a certain parameterization, but using Sashsa' factoring has given me what i needed to finish the problem, just checking over a few more things though – rckrd Oct 2 '12 at 1:56
Factor your coefficients: $$v^4 + 2 v^2 + 1 + u^2 + u^2 v^2 = (1+v^2)(1+u^2+v^2)$$ $$u^4 + 2 u^2 + 1 + v^2 + u^2 v^2 = (1+u^2) (1+u^2+v^2)$$ $$(-2 u v-2 u^2 v-2 u v^3) = - 2u v(1+u^2+v^2)$$ Then cancel common factors of the numerator and the denominator.
After I've calculated $ds^2$ and $da$, the area element, I'm having trouble finding the bounds in which to integrate. – rckrd Oct 2 '12 at 2:05
If you mean to integrate over the hemisphere, then $u$ and $v$ are unconstrained, that integrate over $(u,v) \in \mathbb{R}^2$. – Sasha Oct 2 '12 at 2:28 | crawl-data/CC-MAIN-2016-18/segments/1461860111455.18/warc/CC-MAIN-20160428161511-00118-ip-10-239-7-51.ec2.internal.warc.gz | null |
# Relative Velocity
Topics: Velocity, Relative velocity, Angle Pages: 9 (2834 words) Published: January 21, 2013
Relative Velocity and Riverboat Problems
On occasion objects move within a medium that is moving with respect to an observer. For example, an airplane usually encounters a wind - air that is moving with respect to an observer on the ground below. As another example, a motorboat in a river is moving amidst a river current - water that is moving with respect to an observer on dry land. In such instances as this, the magnitude of the velocity of the moving object (whether it be a plane or a motorboat) with respect to the observer on land will not be the same as the speedometer reading of the vehicle. That is to say, the speedometer on the motorboat might read 20 mi/hr; yet the motorboat might be moving relative to the observer on shore at a speed of 25 mi/hr. Motion is relative to the observer. The observer on land, often named (or misnamed) the "stationary observer" would measure the speed to be different than that of the person on the boat. The observed speed of the boat must always be described relative to who the observer is. To illustrate this principle, consider a plane flying amidst a tailwind. A tailwind is merely a wind that approaches the plane from behind, thus increasing its resulting velocity. If the plane is traveling at a velocity of 100 km/hr with respect to the air, and if the wind velocity is 25 km/hr, then what is the velocity of the plane relative to an observer on the ground below? The resultant velocity of the plane (that is, the result of the wind velocity contributing to the velocity due to the plane's motor) is the vector sum of the velocity of the plane and the velocity of the wind. This resultant velocity is quite easily determined if the wind approaches the plane directly from behind. As shown in the diagram below, the plane travels with a resulting velocity of 125 km/hr relative to the ground. [pic]
If the plane encounters a headwind, the resulting velocity will be less than 100 km/hr. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Suppose a plane traveling with a velocity of 100 km/hr with respect to the air meets a headwind with a velocity of 25 km/hr. In this case, the resultant velocity would be 75 km/hr; this is the velocity of the plane relative to an observer on the ground. This is depicted in the diagram below. [pic]
Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a side wind of 25 km/hr, West. Now what would the resulting velocity of the plane be? This question can be answered in the same manner as the previous questions. The resulting velocity of the plane is the vector sum of the two individual velocities. To determine the resultant velocity, the plane velocity (relative to the air) must be added to the wind velocity. This is the same procedure that was used above for the headwind and the tailwind situations; only now, the resultant is not as easily computed. Since the two vectors to be added - the southward plane velocity and the westward wind velocity - are at right angles to each other, the Pythagorean theorem can be used. This is illustrated in the diagram below. [pic]
In this situation of a side wind, the southward vector can be added to the westward vector using the usual methods of vector addition. The magnitude of the resultant velocity is determined using Pythagorean theorem. The algebraic steps are as follows: (100 km/hr)2 + (25 km/hr)2 = R2
10 000 km2/hr2 + 625 km2/hr2 = R2
10 625 km2/hr2 = R2
SQRT(10 625 km2/hr2) = R
103.1 km/hr = R
The direction of the resulting velocity can be determined using a trigonometric function. Since the plane velocity and the wind velocity form a right triangle when added together in head-to-tail fashion, the angle between the resultant vector and the southward vector can be determined using the sine, cosine, or tangent functions. The tangent function can be used; this is shown below: | crawl-data/CC-MAIN-2018-26/segments/1529267863411.67/warc/CC-MAIN-20180620031000-20180620051000-00124.warc.gz | null |
Polynomials
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# Polynomials - PowerPoint PPT Presentation
Polynomials. The Degree of ax n. If a does not equal 0, the degree of ax n is n . The degree of a nonzero constant is 0. The constant 0 has no defined degree. Definition of a Polynomial in x. A polynomial in x is an algebraic expression of the form
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## PowerPoint Slideshow about ' Polynomials' - rajah-soto
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Presentation Transcript
The Degree of axn
• If a does not equal 0, the degree of axn is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.
Definition of a Polynomial in x
• A polynomial in x is an algebraic expression of the form
• anxn + an-1xn-1 + an-2xn-2 + … + a1n + a0
• where an, an-1, an-2, …, a1 and a0 are real numbers. an= 0, and n is a non-negative integer. The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term.
Text Example
Perform the indicated operations and simplify:
(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)
Solution
(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)
= (-9x3 + 13x3) + (7x2 + 2x2) + (-5x – 8x) + (3 – 6) Group like terms.
= 4x3 + 9x2 – (-13x) + (-3) Combine like terms.
= 4x3 + 9x2 + 13x – 3
Multiplying Polynomials
The product of two monomials is obtained by using properties of exponents. For example,
(-8x6)(5x3) = -8·5x6+3 = -40x9
Furthermore, we can use the distributive property to multiply a monomial and a polynomial that is not a monomial. For example,
3x4(2x3 – 7x + 3) = 3x4 · 2x3 – 3x4 · 7x + 3x4 · 3 = 6x7 – 21x5 + 9x4.
monomial
trinomial
Multiplying Polynomials when Neither is a Monomial
• Multiply each term of one polynomial by each term of the other polynomial. Then combine like terms.
Using the FOIL Method to Multiply Binomials
last
first
(ax + b)(cx + d) = ax · cx + ax · d + b · cx + b · d
Product of
First terms
Product of
Outside terms
Product of
Inside terms
Product of
Last terms
inner
outer
Text Example
Multiply: (3x + 4)(5x – 3).
Text Example
Multiply: (3x + 4)(5x – 3).
Solution
(3x + 4)(5x – 3) = 3x·5x + 3x(-3) + 4(5x) + 4(-3)
= 15x2 – 9x + 20x – 12
= 15x2 + 11x – 12 Combine like terms.
last
first
F
O
I
L
inner
outer
The Product of the Sum and Difference of Two Terms
• The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term.
The Square of a Binomial Sum
• The square of a binomial sum is first term squared plus 2 times the product of the terms plus last term squared.
The Square of a Binomial Difference
• The square of a binomial difference is first term squared minus 2 times the product of the terms plus last term squared.
Special Products
Let A and B represent real numbers, variables, or algebraic expressions.
Special ProductExample
Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B2 (2x + 3)(2x – 3) = (2x) 2 – 32
= 4x2 – 9
Squaring a Binomial
(A + B)2 = A2 + 2AB + B2 (y + 5) 2 = y2 + 2·y·5 + 52
= y2 + 10y + 25
(A – B)2 = A2 – 2AB + B2 (3x – 4) 2 = (3x)2 – 2·3x·4 + 42
= 9x2 – 24x + 16
Cubing a Binomial
(A + B)3 = A3 + 3A2B + 3AB2 + B3 (x + 4)3 = x3 + 3·x2·4 + 3·x·42 + 43
= x3 + 12x2 + 48x + 64
(A – B)3 = A3 – 3A2B – 3AB2 + B3 (x – 2)3 = x3 – 3·x2·2 – 3·x·22 + 23
= x3 – 6x2 – 12x + 8
Text Example
Multiply: a. (x + 4y)(3x – 5y) b. (5x + 3y) 2
• Solution
• We will perform the multiplication in part (a) using the FOIL method. We will multiply in part (b) using the formula for the square of a binomial, (A + B) 2.
• a. (x + 4y)(3x – 5y) Multiply these binomials using the FOIL method.
• = (x)(3x) + (x)(-5y) + (4y)(3x) + (4y)(-5y)
• = 3x2 – 5xy + 12xy – 20y2
• = 3x2 + 7xy – 20y2Combine like terms.
• (5 x + 3y) 2 = (5 x) 2 + 2(5 x)(3y) + (3y) 2 (A + B) 2 = A2 + 2AB + B2
• = 25x2 + 30xy + 9y2
F
O
I
L
Example
• Multiply: (3x + 4)2.
Solution:
( 3x + 4 )2=(3x)2 + (2)(3x) (4) + 42=9x2 + 24x + 16 | crawl-data/CC-MAIN-2017-43/segments/1508187824931.84/warc/CC-MAIN-20171022003552-20171022023552-00212.warc.gz | null |
We would like to know how scientists calculate a planet's mass. Please explain it to us in a way that a 4th grader can understand it.
The only way we can measure a planet's mass is through its gravity. This has been the way Earth's mass was measured, too. (We can't directly probe what's in Earth's interior, but we can measure the gravity on the surface.) Since no human ever visited other planets and measured their gravity on the spot, we usually have to resort to other methods. The most commonly used technique is to observe a body orbiting or passing close to the planet and see how its path is affected by the planet's gravity.
For example, if we see a moon orbiting a planet at certain distance from it, the orbital period of the moon at that particular distance will mainly depend on the planet's mass. The more massive the planet, the more strongly it attracts the moon and faster the moon moves. It is straightforward for astronomers to calculate the planet's mass after we have observed the motion of one of its moons for a while.
Mercury and Venus have no moons, so their exact masses were not known until a few decades ago. Before space flight was developed, the only way to measure their gravity was to see how they affect other planets' orbits. Astronomers would measure very small changes in, say, Earth's orbit, that were caused by the attraction of Venus. These changes are small and it was hard to get the exact mass of Venus by this technique. But once spacecraft were launched to Venus and they flew close to it, scientists could easily measure its mass by tracking how these probes were deflected while passing by Venus. The same technique was used for Mercury when the Mariner 10 spacecraft flew by it in 1974.
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EARLY STAGES OF BREAST CANCER
The term “early breast cancer” refers to stages of breast cancer labeled 0, I, and II.
Cancer cells are present in either the lining of a breast lobule or a duct, but they have not spread to the surrounding fatty tissue. This stage is also called ductal carcinoma in situ, or DCIS.
Cancer has spread from the lobules or ducts to nearby tissue in the breast. At this stage and beyond, breast cancer is considered to be invasive. The tumor is 2 cm or less in diameter (approximately 1 inch or less); the lymph nodes are not involved.
Cancer has spread from the lobules or ducts to nearby tissue in the breast. In this stage, the tumor can range from about 2 cm to greater than 5 cm in diameter (approximately 1 to 2 inches); sometimes the lymph nodes may be involved.
A recurrence is a return of breast cancer. After surgery for early breast cancer, adjuvant, or additional, therapy may be given to reduce the chance of a recurrence.
ADVANCED STAGES OF BREAST CANCER
The term “advanced breast cancer” refers to stages of breast cancer labeled III and IV.
Known as locally advanced cancer; tumor may be larger than 5 cm (2 inches) in diameter, and cancer may or may not have spread to lymph nodes or other tissues near the breast.
Known as metastatic; cancer has spread from the breast and lymph nodes under the arm to other parts of the body, such as bone, liver, lung, or brain.
The presence of hormone receptors in the tumor cells is also important. When these receptors are present, the tumor cells depend on hormones, such as estrogen, for growth. Hormone (either estrogen or progesterone) receptor-positive tumors appear to grow less aggressively than those that are estrogen receptor-negative or progesterone hormone receptor-negative.
Women whose tumors are hormone receptor positive have a lower risk of recurrence than than those who are hormone receptor negative. And, in women with hormone receptor positive tumors, adjuvant hormonal treatment reduces that risk.
BREAST HEALTH FOR PREVENTATIVE CARE
The three-step plan for preventive care. Although breast cancer cannot be prevented at the present time, early detection of problems provides the greatest possibility of successful treatment.
WHAT IS THE THREE-STEP PLAN?
Routine care is the best way to keep you and your breasts healthy. Although detecting breast cancer at its earliest stages is the main goal of routine breast care, other benign conditions, such as fibrocystic breasts, are often discovered through routine care.
STEP 1. BREAST SELF-EXAMINATION (BSE)
A woman should begin practicing breast self-examination by the age of 20 and continue the practice throughout her life — even during pregnancy and after menopause. BSE should be done regularly at the same time every month. Regular BSE teaches you to know how your breasts normally feel so that you can more readily detect any change.
Changes may include:
• Development of a lump
• A discharge other than breast milk
• Swelling of the breast
• Skin irritation or dimpling
• Nipple abnormalities (i.e., pain, redness, scaliness, turning inward)
If you notice any of these changes, see your health care provider as soon as possible for evaluation.
STEP 2. CLINICAL EXAMINATION
A breast examination by a physician or nurse trained to evaluate breast problems should be part of a woman’s physical examination.
The American Cancer Society recommends:
•Between the ages of 20 and 39, women should have a clinical breast examination by a health professional every 3 years.
• After age 40, women should have a breast exam by a health professional every year.
• A physical breast examination by a physician or nurse is very similar to the procedures used for breast self examination. Women who routinely practice BSE will be prepared to ask questions and have their concerns addressed during this time.
STEP 3. MAMMOGRAPHY
Mammography is a low-dose x-ray of the breasts to find changes that may occur. It is the most common imaging technique. Mammography can detect cancer or other problems before a lump becomes large enough to be felt, as well as assist in the diagnosis of other breast problems. However, a biopsy is required to confirm the presence of cancer.
Because when to begin and how often to have mammograms is controversial, talk with your physician about a mammography schedule that is appropriate for you based on your overall health and medical history, risk factors, and personal opinion or preference.
According to the National Cancer Institute, women in their 40s and older should begin having a screening mammogram on a regular basis, every 1 to 2 years. But, the American Cancer Society recommends that by age 40, women should have a screening mammogram every year. (A diagnostic mammogram may be required when a questionable area is found during a screening mammogram.)
MYTHS ABOUT BREAST CANCER
This section outlines some of the common myths and misconceptions about breast cancer. False rumors about breast cancer are becoming more frequent with the increased use of email and the Internet. For example, a recent inaccurate e-mail message was widely circulated stating that the use of antiperspirants is a leading cause of breast cancer. The purpose of this section is to dispel false rumors about what causes breast cancer, how the disease develops, and how different treatment options affect patients.
Myth: Only Women Get Breast Cancer.
Fact: It is estimated that 1,450 men will be diagnosed with breast cancer in 2004 and 470 will die from the disease.
Myth: Only Women with a Family History of Breast Cancer are at Risk.
Fact: While a family history of breast cancer can mean that a woman is at higher than average risk of developing breast cancer, more than 80% of women diagnosed with breast cancer have no identifiable risk factors for the disease.
Myth: Breast Cancer is Mainly a Genetic Disease.
Fact: Only a very small percentage (5%-10%) of breast cancer cases are thought to be due to abnormal genes. Researchers have identified two genes on chromosome 17, BRCA1 (breast cancer gene 1) and BRCA2 (breast cancer gene 2), that may increase breast cancer risk (although more genes that affect breast cancer risk may also exist).
However, only 5% of breast cancer cases are related to mutated BRCA1 or BRCA2 genes. Furthermore, a mutated BRCA gene is only one of the risk factors for developing breast cancer. Other high risk factors include: age, family history, high fat diets, obesity, previous breast biopsy showing benign (non-cancerous) conditions, menstruation beginning at an early age, menstruation continuing past age 50, not having children, having a first child after age 30, etc. Also, up to 80% of women who get breast cancer have no identifiable risk factors.
Myth: Older Women are Less Likely to Get Breast Cancer Than Younger Women.
Fact: As a woman’s age increases, her risk of getting breast cancer also increases. In fact, age is one of the strongest risk factors for developing breast cancer. To help detect breast cancer early, women forty years of age and older should get regular mammograms in addition to a yearly clinical breast examinations (CBE) and monthly breast self-examinations (BSE). Women between the ages of 20 and 40 should also practice monthly breast self-exams and receive physician-performed clinical breast exams at least every three years.
Myth: Breast Cancer is Contagious.
Fact: Cancer is not a communicable disease. Breast cancer is defined as an abnormal increase in breast cells, resulting in a malignant (cancerous) tumor of the breast tissue. Changes in one woman’s cells cannot affect the cells of another woman. Generally accepted risk factors of breast cancer include:
• Family history
• Previous breast biopsy showing benign conditions
• Menstruation beginning at an early age
• Menstruation continuing past age 50
• Not having children
• Having a first child after age 30
• High fat diets
• Mutations of the genes, BRCA1 and BRCA2
Myth: All Breast Lumps are Cancerous.
Fact: In general, 80% of lumps are caused by benign (non-cancerous) changes in the breast. This percentage tends to fluctuate with age. For young women, more than 80% of breast lumps are benign. As a woman ages, her risk for breast cancer increases. The percentage of benign breast lumps in older women may be much lower than in younger women. It is still important for women to report any breast abnormality to their physician, especially if it persists after two or more menstrual cycles.
Myth: A Woman with Lumpy Breasts is at High Risk of Developing Breast Cancer.
Fact: In the past, health care professionals believed women with lumpy breasts were at higher risk for breast cancer. However, this myth has recently been dispelled. Women with lumpy breasts often suffer from a benign (non-cancerous) condition called fibrocystic change.
Symptoms of fibrocystic change in the breast include cysts (accumulated packets of fluid), fibrosis (formation of scar-like connective tissue), lumpiness, areas of thickening, tenderness, or breast pain. One type of rare benign growth, atypical hyperplasia (abnormal increase in the number of breast cells), may increase a woman’s risk of invasive breast cancer. However, only about 3% of breast biopsies reveal atypical hyperplasia.
Myth: Small-Breasted Women Cannot Get Breast Cancer.
Fact: The amount of breast tissue a woman has does not affect her risk of developing breast cancer. Breast size is certainly not a significant risk factor for breast cancer.
Myth: Fibrocystic Change Increases a Woman’s Risk of Developing Breast Cancer.
Fact: Fibrocystic change is a benign (non-cancerous) breast condition and does not increase a woman’s risk of developing breast cancer. However, in some instances, fibrocystic change can make breast cancer more difficult to detect with mammography. This is because the breast density associated with fibrocystic breasts may eclipse breast cancer on a mammogram film. Therefore, it is important that breast self-exams and clinical breast exams also be preformed. In some cases, women with fibrocystic breasts may need additional breast imaging, such as ultrasound, if cancer is suspected but not detectable with mammography.
Myth: Drinking Coffee Increases a woman’s Risk of Developing Breast Cancer.
Fact: Coffee does not cause breast cancer, and in several studies with rats, coffee has been shown to actually prevent cancer. Health care professionals once believed that caffeine caused fibrocystic change (a common non-cancerous breast condition characterized by cysts, lumpiness, tenderness, pain, etc.). Some women find that reducing their caffeine intake by avoiding coffee, tea, chocolate, and soft drinks decreases water retention and breast discomfort. This is a controversial topic among health care professionals, though, since studies linking breast pain and caffeine have been inconsistent.
Myth: Antiperspirants or Antiperspirants/deodorant Combinations are a Leading Cause of Breast Cancer.
Fact: Antiperspirants (or antiperspirant/deodorant combinations) do not cause breast cancer. A false rumor has been broadly circulated claiming that antiperspirants prevent the body from purging dangerous toxins. The message reports that because antiperspirants actually work to stop underarm perspiration (as opposed to regular deodorants that merely provide fragrance), certain toxins become trapped inside the body. These toxins, according to the rumor, are deposited in the lymph nodes below the arms, leading to cell mutations and the development of breast cancer. This link between antiperspirants and breast cancer is completely inaccurate. The body does not, in fact, need to purge toxins from the armpits in the form of perspiration. There are no toxins to purge; sweat is made up of a combination of 99.9% water, sodium, potassium and magnesium. The National Cancer Institute and the U.S. Food and Drug Administration are unaware of any substantial evidence that antiperspirants cause breast cancer.
Myth: Pesticides, Lawn Chemicals, and/or Dry Cleaning Services Cause Breast Cancer.
Fact: A number of small studies over the past few years have shown a possible increased incidence of breast cancer in women who use dry cleaning services or professional lawn services. However, several health care professionals doubt the scientific validity of these studies whose data is often contradicted in larger studies. Similar data linking pesticides to increased incidences of breast cancer have also been inconclusive.
Myth: If a Woman is Diagnosed with Lobular Carcinoma in situ (LCIS), She Will Definitely Develop Breast Cancer.
Fact: Though technically a Stage 0 cancer, most physicians do not consider lobular carcinoma in situ (LCIS; also called lobular neoplasm) to be cancer. However, LCIS is a marker for increased breast cancer risk. Women with LCIS are more likely to develop cancer in either breast later in their lives. LCIS begins in the lobules (the milk-producing glands of the breast) but does not penetrate the lobular walls.
Myth: Breast-Feeding Causes Breast Cancer.
Fact: Breast-feeding does not cause breast cancer. In fact, some preliminary studies reveal that breast-feeding may decrease a woman’s risk of developing breast cancer. However, this data has not been confirmed. Women who breast-feed can still get breast cancer, but they are not at any increased risk compared to women who do not breast-feed.
Myth: Nipple Discharge Indicates Breast Cancer.
Fact: Most nipple discharges do not indicate a cancerous condition. Up to 20% of women may experience spontaneous milky, opalescent, or clear fluid nipple discharge. Up to 60% of women experience nipple discharge during breast self-examination. Usually, if the discharge is clear, milky, yellow, or green, it does not indicate cancer. Bloody or watery nipple discharge is considered abnormal; however, only 10% of abnormal discharges are cancerous. Most bloody discharges are due to non-cancerous papillomas. Women should report any worrisome nipple discharges to their physician for clinical examination. Nipple discharge may be a concern if it is:
• Bloody or watery (serous) with a red, pink, or brown color
• Sticky and clear in color or brown to black in color (opalescent)
• Appears spontaneously without squeezing the nipple
• Persistent on one side only (unilateral)
• A fluid other than breast milk
Myth: Underwire Bras Cause Breast Cancer.
Fact: A book published a few years ago called Dressed to Kill suggested that underwire bras can constrict the body’s lymph node system, causing breast cancer. The authors of the book attributed the high rate of breast cancer in North America (compared to less industrialized countries in the world) to the fact that most North American women wear bras. This link between underwire bras and breast cancer is completely inaccurate. The authors of Dressed to Kill did not take into account any other genetic, environmental, or social factors that could contribute to breast cancer risk (such as age, family history, high fat diet, obesity, not having children, etc.).
Myth: An Injury to the Breast Causes Cancer.
Fact: Injury or trauma to the breast does not cause breast cancer. However, the breast may become bruised or develop a benign (non-cancerous) lump as the result of an injury. Fat necrosis is a rare benign breast condition that occurs when fatty breast tissue swells or becomes tender. When the body attempts to repair the damaged breast tissue, the affected area may sometimes be replaced with firm scar tissue. Fat necrosis may be mistaken as cancer on mammogram; however symptoms of fat necrosis usually subside within a month.
Myth: Oral Contraceptive Pills (Birth Control Pills) Cause Breast Cancer.
Fact: Birth control pills do not cause breast cancer, even after prolonged use (10+ years). Though oral contraceptives do contain small amounts of estrogen and progesterone (hormones often linked with increased risk over time), the amount of these hormones is too small to pose a noteworthy risk. Today, most women are prescribed “low-dose” formulas which contain less than 50 micrograms of estrogen (50% to 100% less estrogen than most birth control pills contained before 1975). Low-dose formulas were developed to ease bothersome side effects of the regular-dose pill such as bloating. In one recent study of 3,383 cases of breast cancer from 1976 to 1992, no overall relationship was noted between the duration of oral contraceptive use and breast cancer risk, even among women who used oral contraceptives for more than 10 years. For women who began taking oral contraceptives after 1975 no significant risk of breast cancer has been noted even among those with a family history of breast cancer. Still, women at high risk for breast cancer should discuss any concerns about oral contraceptives with their physicians.
Myth: The Statistic “One in Eight Women Will Develop Breast Cancer” Means that if Eight Women are Randomly Selected, then One of those Eight Women is Guaranteed to Get Breast Cancer.
Fact: The one-in-eight-women statistic is not a per year estimate. Rather, it is calculated over a lifetime to age ninety-five. If researchers were to follow a large group of girls born today and track them until they became ninety-five years old, then one out of every eight of those girls (approximately 12.5%) would develop breast cancer sometime in her lifetime.
Myth: A Mammogram Prevents Breast Cancer.
Fact: A mammogram cannot prevent breast cancer; however mammography is an excellent tool to screen for and detect the disease at an early stage. Currently, mammography is the only FDA approved exam to screen for breast cancer in asymptomatic women (women who have no symptoms of breast cancer such as a lump). To help detect breast cancer early, women forty years of age and older should have a regular mammogram in addition to a yearly clinical breast examinations (CBE) and monthly breast self-examinations (BSE). Women between the ages of 20 and 40 do not typically need annual screening mammograms unless they have special circumstances (i.e., a strong family history of breast cancer). However women 20-40 years of age should practice monthly breast self-exams and receive clinical breast exams at least every three years.
Myth: A Mammogram Causes Breast Cancer.
Fact: A mammogram is a safe procedure that uses extremely low levels of radiation to create detailed images of the breast. Modern mammography systems typically use only about 0.1 to 0.2 rad dose per x-ray (rad is the scientific unit that measures radiation energy dosage). The MQSA (Mammography Quality Standards Act) was created by the American College of Radiology (ACR) and passed by Congress to mandate rigorous guidelines for x-ray safety during mammography. The MQSA guidelines assure that mammography systems are safe and use the lowest dose of radiation possible. Patients should make sure they are being imaged at an ACR accredited facility using modern mammography systems.
Myth: Mammography is 100% Accurate in Early Breast Cancer Detection.
Fact: Mammography is considered the gold standard for breast cancer detection. However, it is not 100% at detecting breast cancer. Overall, mammography is about 80% effective at detecting breast cancer, when all age groups are considered. However, individual characteristics, such as age, breast density, menopausal status, etc. may affect the accuracy of mammography. For example, sometimes an irregularity goes undetected because surrounding breast tissue is the same density as the irregular tissue. If a patient has a lump or other change and the mammogram is “negative” (interpreted as not suspicious or cancerous), the patient should still pursue that finding with her physician.
Myth: Mammography Always Finds Cancer When it is Curable.
Fact: Mammography is the most accurate screening tool for breast cancer. While annual screening mammograms will detect the vast majority of breast cancers, some cancers are extremely aggressive and can metastasize (spread) to other areas of the body before they are detected by mammogram. In general, breast cancer has a slow rate of growth. It may take six to eight years for a breast cancer developing from one cell to grow to the size of one centimeter. This long growth period allows ample time for aggressive cancers to spread into blood vessels, lymphatic vessels, and beyond the breast. Again, to help detect breast cancer early, when the chances for survival are the greatest, women 20 years of age and older should perform breast self-examination (BSE) every month. Women 20-39 should have a clinical breast examination (CBE) at least every three years in addition to performing monthly BSE. Women 40 and older should practice monthly BSE, have CBE performed by a health care professional every year, and have mammograms every year to two years.
Myth: Breast Cancer Always Presents Itself in the Form of a Lump
Fact: While a breast lump can certainly be a sign of breast cancer (as well as a number of non-cancerous conditions), not all women who are diagnosed with breast cancer will have a noticeable lump. Therefore, women should check for the following warning signs while performing monthly breast self-exams:
• Any new lump or hard knot found in the breast or armpit
• Any lump or thickening that does not shrink or lessen after your next period
• Any change in the size, shape or symmetry of your breast
• A thickening or swelling of the breast
• Any dimpling, puckering or indention in the breast
• Dimpling, skin irritation or other change in the breast skin or nipple
• Redness or scaliness of the nipple or breast skin
• Nipple discharge (fluid coming from your nipples other than breast milk), particularly if the discharge is bloody, clear and sticky, dark or occurs without squeezing your nipple
• Nipple tenderness or pain
• Nipple retraction: turning or drawing inward or pointing in a new direction
• Any breast change that may be cause for concern
While one or more of these changes warrants clinical examination, these changes do not mean that a woman has breast cancer. In addition, breast cancer can be present without any symptoms. For example, screening mammography often detects breast cancer before a lump can be felt. In general, the early breast cancer is diagnosed, the better the chances for successful treatment and survival.
Myth: If a Breast Lump is Painful, Then it is Not Cancerous.
Fact: Up to 10% of breast cancers are associated with pain. However, pain is very rarely the only evidence of a breast tumor. Pain may accompany a breast lump, etc. If a patient has breast pain but physical exams and mammography do not reveal an abnormality, most physicians will not pursue further breast imaging because the likelihood of breast cancer is very small. Breast pain is the third most common non-cancerous breast complaint, and may be caused by a variety of conditions. Bilateral breast pain is less likely to be associated with breast cancer than unilateral breast pain.
Myth: The Best Place to Practice Breast Self-Examination (BSE) is in the Shower.
Fact: BSE can be performed while in the shower. However, wet, soapy hands may make it difficult for a woman to feel the intricacies of her breast. Cold air or water also causes the breasts and nipples to contract. Women over twenty years of age should practice monthly BSE in three positions: lying down, standing up, and standing in front of the mirror (to check for visual breast changes).
Myth: If a Woman is Diagnosed With Breast Cancer, She Will Lose Her Breast.
Fact: Many women who are diagnosed with breast cancer will undergo some type of surgery as part of their treatment. However, breast-conserving therapy (lumpectomy, usually followed by radiation therapy) is becoming common treatment for early stage breast cancers (such as ductal carcinoma in situ (DCIS)). Lumpectomy is the surgical removal of a breast lump and a surrounding margin of normal breast tissue. To date, women with DCIS have chosen equally among lumpectomy and mastectomy (removal of the affected breast), though specific cases may sometimes favor lumpectomy over mastectomy or vice versa. Chemotherapy (the use of anti-cancer drugs) is also being used in some cancer patients to shrink the size of a breast tumor so that a woman may have lumpectomy instead of mastectomy. Recent studies of the drug tamoxifen and other alternative treatments show a growing trend toward less invasive breast cancer treatment.
Myth: Mastectomy Ensures Breast Cancer Will Be Eliminated Forever.
Fact: Mastectomy (removal of the affected breast) does not guarantee that breast cancer will not recur. Some women experience breast cancer recurrence at the site of the mastectomy scar. There is also that possibility that the cancer has spread to the lymph nodes or other areas of the body. Many women who have modified radical mastectomy also undergo axillary lymph node dissection (removal of the underarm lymph nodes) to ensure that the cancer has not spread beyond the breast.
Myth: Women Who Have Prophylactic (Preventive) Mastectomy Will Not Develop Breast Cancer.
Fact: Prophylactic mastectomy is a preventive procedure in which one or both of the breasts are removed in women who are at very high risk for developing breast cancer. The decision to have prophylactic mastectomy should be made carefully after consultation with physicians and family members. Recent research has shown that prophylactic mastectomy can reduce the risk of breast cancer by 90%.
However, some women who are identified to be at high breast cancer never develop disease and thus would not benefit from prophylactic mastectomy. Breast tissue also extends up towards the neck, under the arms, and to the chest wall. A woman is at risk of developing breast cancer as long as breast tissue remains in the body
Myth: Chemotherapy Will Make a Woman’s Hair Fall Out.
Fact: The loss of hair (alopecia) is only one of the temporary side effects of chemotherapy. Hair loss and other side effects of chemotherapy depend on the types of drugs administered, their dosage, and the length of treatment. Some women experience few if any adverse effects from drug treatment. For women who experience alopecia, hair loss usually begins about three weeks after chemotherapy has begun. In most all cases, the hair will regrow after chemotherapy has ended. According to the National Alliance of Breast Cancer Organizations, the early chemotherapy regimen of cyclophosphamide, methotrexate, and flouracil (CMF) causes fewer side effects in most women than other regimens containing Adriamyacin (generic name, doxorubicin).
Myth: Women Who Have Had Breast Cancer in the Past Should Not Become Pregnant.
Fact: Studies show that the hormonal and metabolic changes that occur during pregnancy do not typically pose any significant risk of recurring breast cancer. Additionally, neither the number of pregnancies nor the time lapsed between treatment for breast cancer and pregnancy appear to have any noticeable effect on long-term breast cancer prognosis. Breast cancer survivors who are thinking of becoming pregnant should discuss their medical situation with their physician
BREAST CANCER IN MEN
Breast cancer in men is rare– less than 1% of all breast carcinomas occur in men. The American Cancer Society estimates that in 2004 about 1,450 new cases of invasive breast cancer will be diagnosed among men in the US. The average age at diagnosis is between 60 and 70, although men of all ages can be affected with the disease.
What are risk factors for breast cancer in men?
Risk factors may include:
• Radiation exposure
• Estrogen administration
• Diseases associated with hyperestrogenism, such as cirrhosis or Klinefelter’s syndrome
•Also, there are definite familial tendencies for developing breast cancer:
• An increased incidence is seen in men who have a number of female relatives with breast cancer.
•An increased risk of male breast cancer has been reported in families in which a BRCA2 mutation has been identified.
What is the most common type of breast cancer in men?
Infiltrating ductal cancer is the most common tumor type, but intraductal cancer, inflammatory carcinoma, and Paget’s disease of the nipple have been described as well.
Lobular carcinoma in situ has not been identified in men.
What are the symptoms of breast cancer in men?
The following are the most common symptoms of breast cancer in men. However, each individual may experience symptoms differently. Symptoms may include:
• Breast lumps
• Nipple inversion
• Nipple discharge (sometime bloody)
• A pain or pulling sensation in the breast
The symptoms of breast cancer may resemble other conditions or medical problems. Consult a physician for diagnosis.
What are the similarities to breast cancer in women?
Lymph node involvement and the hematogenous pattern of spread are similar to those found in female breast cancer. The staging system for male breast cancer is identical to the staging system for female breast cancer.
Prognostic factors that have been evaluated include the size of lesion and the presence or absence of lymph node involvement, both of which correlate well with prognosis.
Overall survival is similar to that of women with breast cancer. The impression that male breast cancer has a worse prognosis may stem from the tendency toward diagnosis at a later stage.
What are the treatment options for men with breast cancer?
Specific treatment for male breast cancer will be determined by your physician(s) based on:
• Your overall health and medical history
• Extent of the disease
• Your tolerance for specific medications, procedures, or therapies
• Expectations for the course of the disease
• Your opinion or preference
The primary standard treatment is a modified radical mastectomy, just as it is with female breast cancer. Adjuvant therapy may be considered on the same basis as it is for a woman with breast cancer– since there is no evidence that prognosis is different for men or women. | <urn:uuid:041ebd90-3f66-4135-8219-f6cea1553a7b> | {
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# Fun Math Activities for the 6th Grade Classroom
By Keren Perles
Learning about fractions, perimeter and area, and percentages might seem fascinating to you, but your students may think otherwise. Catch their interest with some of these fun 6th grade math activities.
## Percentages
Percentages tend to bore 6th grade students. After all, what do they care about earning interest and calculating statistics? There are percentages, however, that they can relate to: sales. After all, if they see a pair of designer jeans on sale for 60% off, they have to know whether they can afford them, right?
Tap into this interest by making your own “store." Over the course of a week or so, generously hand out fake money to students who are working well, complete assignments, or take on extra work. At the end of the week, set out different prizes along a table in the classroom. Provide various cheap objects that students might like, such as decorative school supplies, snack foods, or gift certificates for class privileges, and label them with both prices and sale signs. For example, a small notebook might have the price “\$5.00" on it, with a for sale sign that reads “20% off." Have students make their selections on paper and hand them in to you at the end of the period. That night, check their papers to make sure that each student has calculated correctly. The next day, give each student the objects that she calculated correctly.
## Fractions
Are your students struggling with fractions? Helping them visualize the abstract concepts can help. Have them fold a paper into quarters. Then show them how to fold a second paper into eighths. Have them color in one quarter of the first paper, and then ask them to color in an equivalent value on the second paper. It should be easy for them to see that they’ll need to color in two sections of the paper, or 2/8, to get the same value. You can also teach addition and subtraction of fractions this way.
## Perimeter and Area
Figuring out the perimeter and area of a shape can be boring for 6th grade students. Help students use their newfound skills to solve some interesting puzzles with these 6th grade math activities. Provide them with graph paper and encourage them to use it to test out their answers. Have students work in groups to answer some of the following questions:
· What is the largest perimeter a rectangle can have, if its area is 24 cm?
· What is the largest area a rectangle can have, if its perimeter is 20 cm?
· Draw a circle and a square that have approximately the same area. How are the radius of the circle and the length of the square’s side related? | crawl-data/CC-MAIN-2017-30/segments/1500549426169.17/warc/CC-MAIN-20170726142211-20170726162211-00710.warc.gz | null |
About the Hubble Space Telescope
Description of the Hubble
Deployed on April 25, 1990, the Hubble Telescope is a giant observatory aboard a spacecraft. It can make observations of the universe using visible, near-ultraviolet and near-infrared light spectra above the filtering effect of earth's atmosphere. Because of its ability to capture faint light in fine detail and the precision of its observations the Hubble Space Telescope is rapidly expanding astronomers understanding of the cosmos.
About the Planets
Planets to be Observed
The Live from the Hubble Space Telescope observations are targeted for March 1996 when 4 planets - Jupiter, Uranus, Neptune and Pluto - will be in a position for observation by the telescope. The planets Neptune and Pluto were selected (via the Great Planet Debate) as targets for observations by students. | <urn:uuid:168b9358-d8be-48d3-ba94-8fe6faeaa9ff> | {
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Analyze This Part 2!
15 teachers like this lesson
Print Lesson
Objective
The students will be able to use measures of central tendency in real life situations.
Big Idea
This lesson will be connecting mean, median, mode and range to real life. It will allow students to make sense of problems by using previously taught strategies.
DO NOW
10 minutes
The students will be looking at a calculating mean question to get them ready for the day’s lesson. The question involves the amount of cans of soda 6th grade student’s drink in a day. They will be asked to calculate the mean and explain how they got their answer along with deciding if the mean is a good way to describe the data(SMP2) This problem is an informal assessment of how well the students can find the mean of a data set.** In this problem, I’m looking to see if students can calculate the mean and then be able say that the mean is not representative of the data. The mean in this problem is 2.5 and some students drink much more and some much less. The data itself is symmetrical with a small spread.*** I will be watching to see that students use the data value of zero in their mean calculation.
Application of Measures of Central Tendency
60 minutes
The middle part of this lesson will consist of a power point presentation on applying the measures of central tendency and following up with an Around the room. The power point will focus on three concepts: choosing the best description for the data, exploring the affects of an outlier, and applying mean to real life scenarios. Slides are presented so there is one direct instruction with answers and one similar problem for students to grapple with on their own. It might be best to allow students time to talk over and explain their answers with their tablemates.
Following the instruction, the students will be participating in an around the room activity with a partner. The ATR (around the room) will consist of 13 problems for the students to solve. The questions will focus on the measures of central tendency and what they tell us and how they can be affected.
• Have students number their paper from 1 – 13. ( I like to have them fold the paper in half three times as this allows boxes for work space)
• You can group students randomly or have them work with a tablemate or let them choose
• Students will walk around the room and solve the problems in the correct space on their paper. Before moving on they must check answers with their partner.
• Only one set of students at a problem at a time.
Before moving to the final wrap up, have the pairs of students show how they solved one of the around the room problems. This is a great way to check for misunderstandings before moving on to a new concept. Group similar concept problems together to build a common understanding. For example: describing data sets, affects of outliers, and finding missing values.
Closure
10 minutes
The students will be answering the following questions to assess their understanding of the measures of central tendency.
1. Describe a situation in which the mean best describes the data set (Looking for them to say that data values will be similar and contain no outlier)
2. Tell which measure of central tendency must be a data value (mode)
3. Explain how an outlier affects the mean, median, mode and range. An outlier will bring the mean up or down depending on its value. It generally does not affect the median. The mode is affected only if it is the outlier. The range will be affected because the spread of data will be larger. | crawl-data/CC-MAIN-2018-43/segments/1539583514162.67/warc/CC-MAIN-20181021161035-20181021182535-00182.warc.gz | null |
Summary and Analysis
Chapter I - Moral Virtue as a Result of Habits
It has been shown that there are two kinds of virtue — intellectual and moral. Intellectual virtue is the result of learning. Moral virtue, on the other hand, comes about as the result of habit and practice. This shows that the moral virtues are not implanted in man by nature, for nothing created by nature can be made to change its direction or tendency by habit, nor are the moral virtues produced in man against nature. Man is not born either moral or immoral, but he has the capacity to develop moral virtue and this capacity can only be developed through habituation.
The development of moral excellence is not comparable to the development of other human capabilities. All men are endowed with certain faculties by nature. The ability to use these faculties is acquired before they are actually used (e.g., man has the ability to see before he sees, he has the ability to hear before he hears). The moral virtues, though, are acquired only by exercising them, just as skill in the arts and crafts is acquired only through use. For example, just as men become builders by building and harpists by playing the harp, so they become just by performing just actions and temperate by exercising self-control. This view is corroborated by what can be observed in any political system. Legislators seek to make good men of their citizens by making good behavior habitual through good laws. It is success or failure in this area that makes the difference between a good and a bad constitution.
The same factors that produce any excellence or virtue can also destroy it, and this is also true in the arts and crafts. For instance, it is only by playing the harp that a man becomes either a good or bad harpist. If this were not so, there would be no need for teachers and everyone would be born either a good or a bad craftsman. Likewise, it is only by action and by dealing with other men that one is able to become either just or unjust, brave or cowardly, temperate or intemperate.
Thus, it is possible to make this generalization — that characteristics develop from corresponding activities. For this reason we must be certain that our activities are of the right kind, for any variation in them will be reflected in our dispositions. This point underscores the importance of early education, for it makes a great difference whether or not one is inculcated in certain habits from an early age. | <urn:uuid:cbb3ae8c-2741-4a12-ac16-a8b644bfe1ad> | {
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July 28, 2000: Lackluster comet LINEAR (C/1999 S4) unexpectedly threw astronomers a curve. Using the Hubble telescope, researchers were surprised to catch the icy comet in a brief, violent outburst when it blew off a piece of its crust, like a cork popping off a champagne bottle. The eruption, the comet's equivalent of a volcanic explosion (though temperatures are far below freezing, at about minus 100 degrees Fahrenheit in the icy regions of the nucleus or core), spewed a great deal of dust into space. This mist of dust reflected sunlight, dramatically increasing the comet's brightness over several hours. Hubble's sharp vision recorded the entire event and even snapped a picture of the chunk of material jettisoned from the nucleus and floating away along the comet's tail.See the rest:
The orbiting observatory's Space Telescope Imaging Spectrograph tracked the streaking comet for two days, July 5 to 7, capturing a dramatic leap in its brightness [left image]; followed by seeing a wave of newly created dust from the outburst flowing into the coma, a shell of dust surrounding the core [middle image]; and culminating in the discovery of a castoff chunk of material from the nucleus sailing along its tail [the bright dot trailing behind the comet in the picture at right]. The white region represents the brightest part of the coma. The nucleus, which is only about a mile wide, cannot be seen in these images because it's too small for the Hubble telescope to see.
Astronomers list several theories for the eruption. One possible reason is that a particularly volatile region of the core became exposed to sunlight for the first time and vaporized away very suddenly. Another possibility is that a buildup of gas pressure from sublimating ice (a change from ice to gas) trapped just below the comet's surface explosively "blew the lid off" a pancake-shaped layer of crust from its surface. The pressure from sunlight blew the fragment down the tail much like the wind propels a sailboat where it disintegrated into smaller and smaller pieces, eventually becoming too small to see.
Yet another possibility is that the observed fragment is one of the house-sized "cometesimals" that are thought to make up the nucleus. Evidence accumulated during the past decade suggests that comet nuclei are "rubble piles" of loosely held together cometesimals. Perhaps one of the "building blocks" comprising the core broke off and was blown down the tail by a gaseous jet shooting off the comet's surface like a garden hose spray.
Credits: NASA, H. Weaver and P. Feldman (Johns Hopkins University), M. A'Hearn (University of Maryland), C. Arpigny (Liege University), M. Combi (University of Michigan), M. Festou (Observatoire Midi-Pyrenees), and G.-P. Tozzi (Arcetri Observatory) | <urn:uuid:63555449-dd07-468c-9067-fddff554ad85> | {
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Is this general solution for ODE correct?
• andrey21
Based on the conversation, the general solution for the given ODE is x = (Ae^(sin t))/3 and the solution provided by the expert is correct. In summary, the general solution for the ODE dx/dt = 3x^(2) cos t is x = (Ae^(sin t))/3, and the expert has provided a correct solution for making x the subject of the solution.
andrey21
Find the general solution of the following ODE:
dx/dt = 3x^(2) cos t
Make x the subject of the solution.
Heres my solution, is this correct?
dx/dt = 3x^(2) cos t
dx/3x^(2) = cos t dt
Integrating both sides gives:
ln (3x^(2)) = sin t + C
3x^(2) = e^(sin t + C)
3x^(2) = Ae^(sin t)
x^(2) = (Ae^(sin t))/3)
x = SQRT(Ae^(sin t))/3)
Jamiey1988 said:
Heres my solution, is this correct?
dx/dt = 3x^(2) cos t
dx/3x^(2) = cos t dt
Integrating both sides gives:
ln (3x^(2)) = sin t + C
here is where you went wrong,
1/3x2 can be written as x-2/3.
You know that ∫xn dx = xn+1/(n+1) + C for n≠-1
So from what you have said:
∫dx/3x^(2) = -1/3x + C or (-x^(-1)/3) + C
Giving a solution of:
-1/3x + C = sin t
That should be correct.
You can easily check your answer by plugging it back into the original differential equation and seeing if it works.
1. What is a general solution for an ODE?
A general solution for an ODE (ordinary differential equation) is a function that satisfies the differential equation for all possible values of the independent variable. It contains a constant of integration, which allows for an infinite number of possible solutions.
2. How can I check if a general solution for an ODE is correct?
To check if a general solution for an ODE is correct, you can substitute the function into the original differential equation and see if it satisfies the equation for all values of the independent variable. Additionally, you can also take the derivative of the function and verify that it matches the given derivative in the differential equation.
3. What is the difference between a general solution and a particular solution for an ODE?
A general solution is a function that satisfies the differential equation for all possible values of the independent variable, while a particular solution is a specific function that satisfies the equation for a given set of initial conditions. A particular solution can be obtained from a general solution by substituting the initial conditions into the function and solving for the constant of integration.
4. Can there be more than one general solution for an ODE?
Yes, there can be an infinite number of general solutions for an ODE. This is because a general solution contains a constant of integration, which can take on different values and result in different functions that still satisfy the differential equation.
5. Is a general solution for an ODE always unique?
No, a general solution is not always unique. As mentioned before, it can have an infinite number of solutions due to the constant of integration. Additionally, for certain types of differential equations, there may be multiple general solutions that satisfy the equation for different ranges of the independent variable. | crawl-data/CC-MAIN-2024-10/segments/1707947476205.65/warc/CC-MAIN-20240303043351-20240303073351-00021.warc.gz | null |
Tiny biobots made from 3D printing can move 236 micrometers per second from power generated by the beating of the cells.
Scientists from the University of Illinois at Urbana-Champaign have 3D-printed mini biological robots (‘biobots’) that are powered by the heart cells of rats. 3D printing means their design is flexible and adaptable for different purposes.
Popular Mechanics reports that first, a flexible gel scaffold was printed and seeded with the heart cells. This cardiac tissue spread over the hydrogel and the cells were then powered by a liquid food that made them beat and move the biobot in a walking motion. The 7mm biobots currently travel at around 236 micrometers per second, and the team is working on making them faster and more powerful.
Future versions of this biobot will feature skeletal muscle (which is more controllable), incorporate neurons that enable them to detect and combat toxins, and could have two legs to move more freely. Check out the video below to see the movement of the biobots: | <urn:uuid:41ba403b-8c5e-42d0-b68d-fd28ef4a7958> | {
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Scientists Say They’ve Finally Identified the Origin of the Black Death
The source of the Black Death — the inspiration behind numerous black metal songs — is said to have been finally pinpointed by European researchers, according to a report this week in the U.K.'s Metro. The Black Death was the 14th century bubonic plague that's still the deadliest pandemic recorded in human history.
But the place where it first emerged has been disputed by scientists for over six centuries.
Now, perhaps ground zero for the Black Death has finally been located. While the search may seem archaic, the subject holds vast importance for humanity. Especially since the world is currently enduring another deadly pandemic, COVID-19, which has killed over a million Americans since March 2020, according to a New York Times database.
The Black Death is estimated to have killed 75 to 200 million across Africa, Asia and Europe from 1346 to 1353. While most scientists have long believed the virus responsible for it originated somewhere in Asia, it has never been conclusively pinned down.
But the new research team from Germany's Max Planck Institute for Biological Cybernetics and Scotland's University of Stirling said they have found the origin point. They concluded that deaths in an earlier 14th century outbreak in modern Kyrgyzstan were due to strains of the same bacterium, Yersinia pestis, that ultimately created the pathogens found in the Black Death.
"It is like finding the place where all the strains come together, like with coronavirus where we have Alpha, Delta, Omicron all coming from this strain in Wuhan, [China]," German palaeogeneticist Johannes Krause, one of the researchers on the study, explained to Nature.
The researchers found their answer by analyzing DNA (ancient DNA or "aDNA") from the teeth of skeletons from cemeteries in the Tian Shan region of Kyrgyzstan. They did so after identifying a spike in burials in the area in 1338 and 1339. From there, they located the bacterium.
"Our study puts to rest one of the biggest and most fascinating questions in history and determines when and where the single most notorious and infamous killer of humans began," a fellow researcher added.
The Black Death is also known as the Great Mortality, the Pestilence or just the Plague. It is estimated to have killed around 30 to 60 percent of the European population and around a third of the population of the Middle East. It was part of a crisis of the Middle Ages that followed the Great Famine of 1315–1317.
Above photo: Protective clothing for 17th century plague doctors. | <urn:uuid:7f11ac0b-0d4d-4651-8164-9f196286e3c2> | {
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The lower of the conscious species possess only the faculty of sensation, which is sufficient to direct their actions and provide for their needs. A sensation is produced by the automatic reaction of a sense organ to a stimulus from the outside world; it lasts for the duration of the immediate moment, as long as the stimulus lasts and no longer. Sensations are an automatic response, an automatic form of knowledge, which a consciousness can neither seek nor evade. An organism that possesses only the faculty of sensation is guided by the pleasure-pain mechanism of its body . . .
The higher organisms possess a much more potent form of consciousness: they possess the faculty of retaining sensations, which is the faculty of perception. A “perception” is a group of sensations automatically retained and integrated by the brain of a living organism, which gives it the ability to be aware, not of single stimuli, but of entities, of things. An animal is guided, not merely by immediate sensations, but by percepts. Its actions are not single, discrete responses to single, separate stimuli, but are directed by an integrated awareness of the perceptual reality confronting it.
Although, chronologically, man’s consciousness develops in three stages: the stage of sensations, the perceptual, the conceptual—epistemologically, the base of all of man’s knowledge is the perceptual stage.
Sensations, as such, are not retained in man’s memory, nor is man able to experience a pure isolated sensation. As far as can be ascertained, an infant’s sensory experience is an undifferentiated chaos. Discriminated awareness begins on the level of percepts . . .
Percepts, not sensations, are the given, the self-evident. The knowledge of sensations as components of percepts is not direct, it is acquired by man much later: it is a scientific, conceptual discovery . . . .
(It may be supposed that the concept “existent” is implicit even on the level of sensations—if and to the extent that a consciousness is able to discriminate on that level. A sensation is a sensation of something, as distinguished from the nothing of the preceding and succeeding moments. A sensation does not tell man what exists, but only that it exists.)
Sensations are the primary material of consciousness and, therefore, cannot be communicated by means of the material which is derived from them. The existential causes of sensations can be described and defined in conceptual terms (e.g., the wavelengths of light and the structure of the human eye, which produce the sensations of color), but one cannot communicate what color is like, to a person who is born blind. To define the meaning of the concept “blue,” for instance, one must point to some blue objects to signify, in effect: “I mean this.” Such an identification of a concept is known as an “ostensive definition.” | <urn:uuid:5f56ecd4-b04e-4366-af85-a271a3595014> | {
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# Three Million Divided by 99
Welcome to three million divided by 99, our post which explains the division of three million by ninety-nine to you. The numeral three million (3000000) is called the numerator or dividend, and the number 99 is called the denominator or divisor. The quotient of three million and 99, the ratio of three million and 99, as well as the fraction of three million and 99 all mean the same: three million divided by ninety-nine, commonly written as 3000000/99. Read on to find the result of this division in various notations, along with its properties.
## What is Three Million Divided by 99?
We provide you with the result of the division of three million by 99 straightaway:
three million divided by 99 = 30303.03
The result of three million divided by 99 is a non-terminating, repeating decimal. The repeating pattern above, 03, is called repetend, and denoted overlined with a vinculum.
The notation in parentheses is also common: 30303.(03): However, in daily use it’s likely you come across the reptend indicated as ellipsis: 30303.03… .
• three million divided by 99 in decimal = 30303.03
• three million divided by 99 in fraction = 3000000/99
• three million divided by 99 in percentage = 3030303.03030303%
Note that you may use our state-of-the-art calculator below to obtain the quotient of any two numerals, integers or decimals, including three million and ninety-nine, of course. Repetends, if any, are denoted in ().
The conversion is done automatically once the nominator, e.g. three million, and the denominator, e.g. ninety-nine, have been inserted in decimal notation. No need to press the button, unless you want to start over.
The Result is...
Give it a try now with a similar division by 99.
In the next section of this post you can find the frequently asked questions in the context of three million over ninety-nine, followed by the summary of our information.
## Three Million Divided by Ninety-Nine
You already know what three million divided by 99 is, but you may also be interested in learning what other visitors have been searching for when coming to this page. The FAQs include, for example:
• What is the quotient of three million and99?
• How much is three million divided by 99?
• What does three million divided by 99 equal?
If you have read our article up to this line, then we take it for granted that you can answer these FAQs and similar questions about the ratio of three million and 99.
Observe that you may also locate many calculations such as three million ÷ 99 using the search form in the sidebar. The result page lists all entries which are relevant to your query.
Give the search box a go now, inserting, for instance, three million divided by ninety-nine, or what’s three million over 99 in decimal, just to name a few potential search terms.
Further information, such as how to solve the division of three million by ninety-nine, can be found on our home page, along with links to further readings.
To sum up, three million over 99 = 30303.(03). The indefinitely repeating sequence of this decimal is 03.
For questions and comments about the division of three million by 99 fill in the comment form at the bottom, or get in touch by email using a meaningful subject line.
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Purchase Solution
# Functions and infinity
Not what you're looking for?
The reason why polynomials are so important is that there is a theorem from Analysis that says that any continuous function defined on an interval of the real line can be approximated arbitrarily closely by a polynomial. So polynomials are useful to ¿model¿ any kind of function on a closed interval. However, polynomials ¿get wild¿ at infinity, so they don¿t work well to try to extrapolate an arbitrary function past the closed interval in which it is being approximated by the polynomial.
A rational function is a function which is a ratio of two polynomials, one polynomial in the numerator and another one in the denominator. Rational functions are also used to model an arbitrary function, and for many purposes they have better behavior. If the rational function is a ratio of two polynomials of the form p(x)/q(x), and the order of the two polynomials is np and nq, try to give a qualitative description of the behavior of this rational function. What happens to the rational function in the cases np > nq, np = nq, and np < nq as x goes to plus or minus infinity (compare with the case of a polynomial)? If an arbitrary function f(x) goes to zero at plus and minus infinity, what kind of rational function would be best to model this function?
##### Solution Summary
This provides explanations of what happens to certain functions as they approach infinity.
##### Solution Preview
If the rational function is a ratio of two polynomials of the form p(x)/q(x), and the order of the two polynomials is np and nq, try to give a qualitative description of the behavior of this rational function. What happens to the rational function in the cases np > nq, np = nq, and np < nq as x goes to plus or minus infinity (compare with the case of a polynomial)?
CASE1
When order of numerator is greater than denominator i.e. np > nq, the numerator grows at a rate that is much higher than the denominator. Hence, the rational function will tend to + infinity or - infinity depending on the signs.
For example,
let p(x) = 3x^2 + 2 where ^ means power.
let q(x) = x + 1
np =2 amd nq = 1 here.
As x tends to + infinity 3x^2 + 2 grows much faster in magnitude than x + 1. Therefore, the ratio tends to infinity.
In mathematical terms, we can see that 3x^2 + 2/(x + 1) = [3x + 2/x]/(1 + 1/x)
As x tends to infinity, 1/x and 2/x tends to 0. Hence p(x)/q(x) lim x -> infinity = 3x which tends to infinity as x becomes
large on postive side.
If x tends to - infinity, then the numerator tends to + infinity since square is always positive. On the otherhand, x + 1 tends to - ...
##### Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
##### Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
##### Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts | crawl-data/CC-MAIN-2024-33/segments/1722640694449.36/warc/CC-MAIN-20240807111957-20240807141957-00571.warc.gz | null |
Education is an essential tool in our life. It is encompassed critical skills such as decision-making, mental dexterity, logical thinking and problem-solving. People face challenges in their personal and professional life that require key elements to solve them. The problems and challenges may vary in personal life but the design of solving depends upon how well-educated and self-awareness the person is actualized. A good education system in a country helps to measure and determine the working force of a country (Elstad, p.94-110). It helps to measure the invention and innovation that increases the industrialization of a nation.
For instance, the education system enhances the absorption of its students into the external and internal world. It promotes employment, earning, poverty and health reduction among other disparities in society. It also compares the relationship between its employees and functional products in innovation and invention globally thus drawing a broader conclusion of a good educational system. Should the US education system be changed? The US education system should change in comparison with that Iceland’s structure. Generally, the US education system and structure are ranked to be the best in terms of offering inventive and innovative knowledge to its students across the world.
However, the primary and upper section needs to be enhanced and advanced to increase the numbers of children and students of all communities attending basic education to increase chances of opportunities. The US education system is not single as other nations have, the reason being of several districts in the country exercise their regulation and duties to ensure efficient and effective education in the country.
Despite the many educational systems, the US faces many challenges that cannot afford to change education system; some of the challenges include many districts with different policies regulations and frameworks, lack of funding for education, shortage of trained and professional teachers and challenges faced by disparities among racial groups causing a high rate of dropout of some of the students.
The many districts with different policies and structural frameworks in the US mitigate the chances of changing the educational system. For instance, the working environment of a teacher in New York varies from that of Mississippi in the country thus causing a difference in the content offered to the students. Changing the education system in this manner will result in a fall in the economic state of each state in the country. It also leads to a difference in the earning that will increase the minimum wage for the teachers in each state causing turbulence to other economic activities of the country. Unlike in Iceland where the national and local government regulate the education system of the country which result in high performance for the students because a comparison of the performance of each teacher is analyzed.
Lack of funding for change in the US education system is another faced challenge that hinders the development of a new system. The funding of the US education system is estimated to be $1.3 trillion which comes from the national and the local government while the federal government amounts to $ 260 billion in the support of the educational program. The funding of the education program is estimated to be 7.2% of the US GDP. The new educational system will require a higher budget than that of the current state thus leading to the failure of the other sector of the economy. For instance, an additional amount on the educational budget will reduce appropriation on other sectors of the economy such as technology and agriculture causing inflation in the country. Unlike Iceland where the system of education starts funding from the preschool centres. The government of Iceland undertake the responsibility at all levels of education and hence creates and drafts a set of policies that regulate its members across the country(Goff et al, p.153-170). Education is taken as the bedrock of the country’s economy hence many policies and guidelines are passed to enhance its education system globally.
However, changing the US education system could influence great change in the economy. With the available resources; educational stakeholders can create and design sets of policies that allow education to all levels of races despite the disparities across the country to enhance the logic and mentality agility of all children in the state(Merz-Atalik & Kerstin, p.9-34). Educational stakeholders should ensure resource equality and in a just manner to provide opportunities for all thus driving the economy. The pressure for change will increase innovation and invention in the industries, which increases chances of employment, and health and reduces poverty across society.
In conclusion, the educational system helps to equip learners with critical knowledge essential for making profound decision-making. It also helps to increase logical thinking and problem-solving skills on professional and personal levels. The educational system of the US should remain neutral because it requires more funding and resources that will interfere with other economic activities of the countries resulting in inflation.
Elstad, Eyvind. “The Evolution of Extended Universal Compulsory Schooling in Sweden, Norway and Denmark: Policy Borrowing and Path-dependent Processes.” Nordic Studies in Education 43.1 (2023): 94-110.
Goff, Kerby, Eric Silver, and Inga Dora Sigfusdottir. “Academic Orientation as a Function of Moral Fit: The Role of Individualizing Morality.” Sociology of Education 95.2 (2022): 153-170.
Merz-Atalik, Kerstin. “Canada as a “Driving Force” for Inclusion Activists in European Countries? 1: Comparative Perspectives on Inclusive Education in Europe and Canada.” European Perspectives on Inclusive Education in Canada. Routledge, 2022. 9-34.
Essay On Restitution Essay Sample For College
Restitution denotes a lawful act that entails returning something to a victim which was unlawfully taken or compensating them for harm caused. In criminology, restitution is ordered as a part of a plea or sentence agreement where the offenders compensate the victims for the damage they caused (Martin & Fowle, 2020). Punishing offenders due to their unjust acts fails to address the core causes as well as social circumstances that prompted criminality, and thus punishments need to be lenient and integrate more rehabilitative approaches such as restitution. Restitution strives to make victims whole again by reinstating their position before the crime happened. Besides, when offenders fulfill their restitution obligations, they experience greater self-worth feeling as they recognize the price of their criminal conduct. More so, restitution holds a deterrent influence since offenders attempt to avoid future delinquency and guarantee that victims’ status will be restored after damage.
Remarkably, restitution takes various forms, like community services, returning taken property, and monetary payments. Community service restitution seems to be common when adult offenders are involved since it avoids offenders’ contact with their victims. This seems easy as the offender is not required to get a job to pay the victim, and monitoring the offender’s improvement gets easier. Monetary restitution refers to the reimbursement a victim receives to cover losses instigated by the misconduct (Martin & Fowle, 2020). Also, the offender may be forced to return stolen assets or property to the victim. These restitutions are under two classes indirect and direct type. Direct restitution entails paying the victims directly, particularly through money for damages caused (Martin & Fowle, 2020). On the other hand, indirect restitution demands other work or community services that benefit society as a whole (Martin & Fowle, 2020). Besides, indirect restitution is ideal when direct compensation will not be adequate or possible for the victim. This paper highlights different restitution types, associated problems, its collection, improvement areas, and new policies or programs that could help reestablish victims.
Restitution, although an alternative punishment, still comes with its challenges. Among the main issues in restitution is that some wrongdoers cannot pay the requested sum due to restricted financial resources or are reluctant to commit due to a lack of repentance or meanness (Martin & Fowle, 2020). Besides, this leaves victims with deep hollows since they are not compensated for the damages and harm they suffered. Again, the restitution collection process seems complicated and lengthy (Paik, 2020). Victims could be needed to maneuver intricate legal systems; after all, they may fail to receive compensation for years. Besides, victims may lack the know-how to collect restitution or may have limited access to legal support and resources to assist them.
Restitution is usually ordered via a court directive. Notably, the court determines the restitution amount owed and orders the wrongdoer to recompense the victim (Bawono, 2021). When the offender fails, legal consequences may follow, such as imprisonment or fines. Typically, restitution is collected through various methods like tax refunds, property liens, or wage garnishment. Even though these approaches may seem effective, much more can be done to streamline the process.
One area that can be improved in this process is ensuring that victims have it easier when collecting their compensation. Besides, this could entail offering more support and resources to victims in getting their payments and smoothening the legal course to ensure victims are not subjected to burdensome processes (Bawono, 2021). For instance, technology utilization can help streamline this restitution process. Digitized wage garnishment and online payments can make the process less burdensome and more efficient for offenders and victims. Also, jurisdictions need to implement a restitution fund for victims, which can assist in compensating the victims when offenders are not in a position to pay (Bawono, 2021). Such a scheme can ensure victims receive something irrespective of the wrongdoer’s financial status. In addition, during sentencing, the importance of restitution should be emphasized, at the courts should take up the role of following if the victim is compensated rather than leaving them alone to pursue their compensation.
New Policies and Programs
Victims have the right to be compensated for damages they have suffered from crimes, and this calls for restitution policies and programs to help victims. Victims compensation fund would be an ideal program to help victims get their payback (Martin & Fowle, 2020). Notably, this fund can help assist crime victims by offering financial support, including lost wages, costs incurred from the crime, or medical bill compensation. Besides, this fund can be publicly financed through taxes, and it would assist victims in moving on with their lives normally and recovering from crime trauma. Education or training programs in which offenders earn can help compensate victims (Bawono, 2021). This would ensure offenders become productive society members and victims receive something to nurse the damages sustained.
Restitution seems an essential component in criminal justice since it holds offenders liable for their crimes and helps in compensating victims for damage incurred. Even though there are some problems with this approach, some areas can be improved for it to be effective. By adopting new policies and programs and leveraging technology, victims can be sure they will receive restitution which they are eligible to. Besides, through restitution, offenders can be seen as responsible society members. Restitution aims to ensure that the justice system becomes just and fair.
Bawono, B. T. (2021). Restitution Rights As A Construction Of Justice Referring To The Law On Protection Of Witnesses And Victims. International Journal of Law Reconstruction, 5(1), 25-36.
Martin, K. D., & Fowle, M. Z. (2020). Restitution without Restoration? Exploring the Gap between the Perception and Implementation of Restitution. Sociological Perspectives, 63(6), 1015-1037.
Paik, L. (2020). Reflection on the Rhetoric and Realities of Restitution. UCLA Criminal Justice Law Review, 4(1).
Safety Policy And Safety Culture University Essay Example
Safety should be the priority for any organization in the aviation industry (Adjekum & Tous, 2020). Therefore, the issue of safety should be addressed with utmost keenness and caution because the results of a failed safety system could be catastrophic and costly to the organization. For this reason, organizations in the aviation industry have laid down policies and regulations that govern the safety of staff and passengers traveling on these airlines. These rules dictate the behavior of each employee towards safety management and enforce compliance with set rules. In addition, organizations adopt a safety culture that directs employees on how to conduct themselves when dealing with threats to safety and raises their commitment to safety management. Notably, a blend of well-designed safety policies and a well-developed safety culture is the foundation of an effective safety management system.
Safety Policy in aviation is a statement of the approach an organization has taken in ensuring safety by defining all safety objectives as well as the responsibilities and accountabilities of the employees and management in achieving acceptable safety levels (Pidgeon & Leary, 2017). Primarily, safety policies are seen as a fundamental aspect of achieving safety in aviation. The safety policy defines the organizational commitment to achieving safety and the steps taken to ensure the policies are followed across the organization. According to ICAO Safety Management Manual, it is the safety policy that should express how the management team is committed to safety management.
Safety responsibility is based on the principle of safety responsibility that dictates that all staff must follow all the rules governing safety in the organization. In addition, safety planning is an important activity through which the organization sets its safety targets and lays down strategies to be followed in achieving set targets. The safety department drafts the safety policies and communicates them to all employees in the organization. Each employee is expected to behave in accordance with the safety policies of their organization, failure to which they face disciplinary action. Notably, the safety department reports periodically on the progress and achievements in safety management and writes a comprehensive report about it. From the report findings, they review the safety policies and make changes where necessary. These policies enable the organization to keep bettering its safety and building a good reputation for itself.
Safety culture in aviation refers to practices and norms of the organization that manage and reduce the risk of accidents and other safety threats. Precisely, safety culture is defined by how people in the organization conduct themselves and their commitment to achieving safety (Adjekum & Tous, 2020). Developing an effective safety culture within an organization is a continuous process that requires the effort of everyone to identify hazards that could compromise safety and implement solutions early in advance. It also involves creating awareness through training on safety practices and the role each person should play in ensuring safety in operations. Furthermore, maintaining a positive safety culture calls for proper reporting on safety issues from all levels of the organization to aid in safety management and planning.
A safety culture of an organization is a reflection of the beliefs, values, and attitudes that individuals in an organization hold regarding safety. It is the safety culture that sets the tone for how safety management is practiced, managed, and monitored. Notably, developing an extensive and effective safety culture is significant in aviation because of the risky nature of operations in the industry. Therefore, there is a need to develop a positive safety culture across organizations in the aviation industry to identify and prepare for risks whose results could be catastrophic if not addressed early. The only way the management can achieve this is by ensuring everyone becomes committed to identifying, reporting, and addressing safety concerns as soon as they arise.
Relationship between safety policies and Culture
Both safety policies and safety vulture are important aspects of safety management in aviation as they complement each other to ensure safety is achieved. However, the two are different in how each is implemented and developed in the organization. Notably, safety policies are formalized rules and procedures laid down to guide safe operations in aviation, while a safety culture consists of all the safety-related attitudes and behaviors that are developed collectively among all members of the organization. It is the safety policies that lay a basis for safety management in aviation. This is achieved by defining all the roles, duties, and responsibilities of employees in the risk management process. Through these policies, the risk is identified, accessed, and effective mitigation measures are taken.
However, the success of safety policies depends mainly on the existence of a good safety culture in the organization. Safety policies alone cannot be much effective in achieving exemplary safety performance but require complimenting with a positive safety culture. Unlike the safety policies, the safety culture requires a voluntary commitment to improving safety through open communication and reporting. It develops a sense of accountability among all employees towards risk reduction, making them more committed to safety management in their daily activities. Through the development of a positive safety culture, employees are motivated to be at the forefront in reporting safety concerns in the course of their duties and even suggest possible solutions to safety threats.
Role of Accountable Executive in Safety Management
The Accountable Executive so critical in implementing effective safety management systems in the aviation industry. Primarily, he is in charge of making sure that safety measures are followed, and safety regulations are adhered to. He does this by establishing a solid safety policy and setting the organization’s objectives regarding safety management. Notably, the policies should be in agreement with the long-term goals of the organization (Pidgeon & Leary, 2017). Additionally, he follows up to ensure the policies and objectives are well communicated to all employees. Another role he performs is to monitor how effective the safety system is. To achieve this, all hazards are identified, and controls are implemented to mitigate threats. Throughout the process, the Accountable Executive ensures accurate data is recorded at every step, and safety is monitored frequently. The data collected is used later to establish trends in risks and ways to improve safety in the future.
Another role of the Accountable Executive is making sure that all the safety regulations are complied with and followed to the letter. Notably, the internal regulatory policies should align with the general standards set by regulators in aviation, such as FAA in the U.S. (Turner, 2019). Besides, he must ensure that internal safety policies are updated to align with changes that happen over time to avoid obsolescence. In addition, it is the role of the Accountable Executive to allocate resources needed to design and implement an effective safety management system, such as training costs. Ultimately, after implementing and monitoring the safety system, it is the role of the Accountable Executive to review the effectiveness of the systems and make adjustments where necessary.
The foundation of an effective safety management system is indeed the establishment of sound safety policies and cultures. Notably, the first step in ensuring safety is altering employee behavior to make them act in accordance with the set safety regulations. However, safety culture is instrumental in influencing employee behavior toward safety. The Accountable Executive should therefore strive to implement policies and cultures that promote safety in organizations in the aviation industry.
Adjekum, D. K., & Tous, M. F. (2020). Assessing the relationship between organizational management factors and a resilient safety culture in a collegiate aviation program with Safety Management Systems (SMS). Safety science, 131, 104909.
Ellis, K. K., Krois, P., Koelling, J., Prinzel, L. J., Davies, M., & Mah, R. (2021). A Concept of Operations (ConOps) of an in-time aviation safety management system (IASMS) for Advanced Air Mobility (AAM). In AIAA Scitech 2021 Forum (p. 1978).
Pidgeon, N., & O’Leary, M. (2017). Organizational safety culture: Implications for aviation practice. In Aviation psychology in practice (pp. 21-43). Routledge.
Turner, B. A. (2019). The development of a safety culture. In Risk Management (pp. 397-399). Routledge. | <urn:uuid:af278779-b14e-4170-9b15-41239dd02752> | {
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MULTIPLE CHOICE In what order should you perform the operations in the
# Multiple choice in what order should you perform the
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39. MULTIPLE CHOICEIn what order should you perform the operations in the expression 4 ×3 12 ÷2 +5? (Section 1.3)A×, , ÷, +B×, ÷, , +C×, ÷, +, D×, + , , ÷ Room 1Room 2adjoining wall4
36 Chapter 1 Numerical Expressions and Factors How can you find the least common multiple of two numbers? Least Common Multiple 1.6 Work with a partner. Using the first several multiples of each number, copy and complete the Venn diagram. Identify any common multiplesof the two numbers.a. 8 and 12Multiplesof 8Multiplesof 12b. 4 and 14Multiplesof 4Multiplesof 14c. 10 and 15Multiplesof 10Multiplesof 15d. 20 and 35Multiplesof 20Multiplesof 35e. Look at the Venn diagrams in parts (a)–(d). Explain how to identify the least common multiple of each pair of numbers. Then circle it in each diagram.Identifying Common MultiplesCommon MultiplesIn this lesson, you willuse diagrams to identify common multiples.find least common multiples. ACTIVITY: 1
Section 1.6 Least Common Multiple 37 Work with a partner. a. Write the prime factorizations of 8 and 12. Use the results to complete the Venn diagram.b. Repeat part (a) for the remaining number pairs in Activity 1.c. STRUCTURECompare the numbers from each section of the Venn diagrams to your results in Activity 1. What conjecture can you make about the relationship between these numbers and your results in Activity 1?ACTIVITY: Interpreting a Venn Diagram of Prime Factors3. IN YOUR OWN WORDSHow can you find the least common multiple of two numbers? Give examples to support your explanation. 2 4. The Venn diagram shows the prime factors of two numbers.232235Use the diagram to do the following tasks.a. Identify the two numbers.b. Find the greatest common factor.c. Find the least common multiple.5. A student writes the prime factorizations of 8 and 12 in a table as shown. She claims she can use the table to find the greatest common factor and the least common multiple of 8 and 12. How is this possible?8 =12 =2232226. Can you think of another way to find the least common multiple of two or more numbers? Explain.Use what you learned about least common multiples to complete Exercises 3–5 on page 40. Primefactorsof 8Primefactorsof 12 Construct ArgumentsHow can you use diagrams to support your explanation? Math Practice
Lesson 1.6 38 Chapter 1 Numerical Expressions and Factors Multiples that are shared by two or more numbers are called common multiples . The least of the common multiples is called the least common multiple (LCM). You can find the LCM of two or more numbers by listing multiples or using prime factors. | crawl-data/CC-MAIN-2021-31/segments/1627046151699.95/warc/CC-MAIN-20210725143345-20210725173345-00675.warc.gz | null |
# Find integral dx / (x + 1) (x + 2)
integration of log x | integration by parts
integration of log x | integration by parts
Example 11 – Chapter 7 Class 12 Integrals
Last updated at May 29, 2023 by Teachoo
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Example 11 Find + 1 + 2 Using partial functions 1 ( + 1)( + 2) = A + 1 + B + 2 1 = (x + 2)A + (x + 1)B 1 = x (A + B) + 2A + B Thus, B = A = 1 Thus our equation becomes, + 1 ( + 2) = 1 + 1 1 + 2 = log +1 log +2 + C = log + + + C
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The brains of autistic children have far more neurons in the prefrontal cortex than the brains of kids without autism, finds a new study that could advance research into the disorder. "For the first time, we have the potential to understand why autism gets started," said study author Eric Courchesne, a professor of neurosciences at the University of California, San Diego School of Medicine and director of the Autism Center of Excellence. "Creating brains cells and the correct number of brain cells is absolutely fundamental to building the brain," said Courchesne. "If there is an excess number of neurons, there must be a negative consequence to that in the way the brain gets wired or organized." In this small, preliminary study, the researchers examined postmortem brain tissue from seven boys with autism and six boys without autism who were aged 2 to 16 when they died. The autistic children had on average 67 percent more neurons — a type of brain cell and a fundamental building block of the nervous system — than boys without autism of a similar age. Specifically, they found autistic children had 79 percent more neurons in the dorsolateral prefrontal cortex and 29 percent more in the mesial prefrontal cortex than other kids. The prefrontal cortex is key to complex thoughts and behaviors, including language, social behavior and decision-making. The dorsolateral prefrontal cortex is closely linked with "executive function," including planning, reasoning and "very high level cognition," said Lizabeth Romanski, an associate professor of neurobiology and anatomy at the University of Rochester Medical Center, who was not involved with the research. The mesial prefrontal cortex is thought to be important to social and other behavior and emotions. While typically developing kids had about 1.16 billion neurons in the prefrontal cortex, autistic children had about 1.94 billion. The study is published in the Nov. 9 issue of the Journal of the American Medical Association. Autism is a neurodevelopment disorder characterized by problems with social interaction, communication and restricted interests and behaviors. An estimated one in 110 U.S. children — many more boys than girls — has the disorder, according to the U.S. Centers for Disease Control and Prevention. Neurons are generated only during fetal development, specifically between 10 weeks and 20 weeks, Courchesne said. While neurons continue to grow in size during childhood, and brain connections get built and pruned, the number of neurons remains constant from birth. That means that whatever goes wrong in autism starts in utero, which should help focus researchers looking for its causes or triggers, including specific genes or prenatal exposures. "Now let's find out what genes or what in-utero, non-genetic conditions lead to an excess number of neurons," he said. Prior research has also documented "brain overgrowth" in autistic children, but those studies were done by measuring brain circumference or MRIs, experts said. In this research, researchers were able to be more specific by counting brain cells in the prefrontal region. "This very nicely builds on previous research and tries to explain why this increase in brain size might be, and what they find is it's because of an increased number of neurons," Romanski said. After a period of proliferation during the second trimester, neurons are also "pruned," meaning that they undergo a planned cell death. "This pruning process, the dying off of cells, is a very important part of brain development," Romanski said. One question that needs to be explored is whether autistic brains generate more neurons, or if they have a malfunctioning "pruning" process, she said. Nicholas Lange, an associate professor of psychiatry and biostatistics at Harvard Medical School in Boston, cautioned that the study was small. He also said more needs to be learned, including whether excess neurons in the prefrontal cortex occur only in autism or in other developmental conditions, or even in any typically developing kids, as well. Some of the kids with autism had many extra neurons, but not all had brains out of the normal range for weight, as would be expected. "The relationship between increased neuron count, brain overgrowth, and increased brain weight in autism is complex," Lange wrote in accompanying editorial. Conducting postmortem brain tissue studies is a lengthy process because there are few brains available to study, Courchesne said. Eight of the 13 children whose brains were studied had drowned. "So very seldom do people at that moment make the decision to donate their child's brain tissue for research, and in the absence of brain tissue for research, it goes very slowly," he said. | <urn:uuid:6ce449d3-2a5f-4792-91c0-3237012a7a02> | {
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Human evolution, the process by which human beings developed on Earth from now-extinct primates. Viewed zoologically, we humans are Homo sapiens, a culture-bearing, upright-walking species that lives on the ground and very likely first evolved in Africa about 315,000 years ago. We are now the only living members of what many zoologists refer to as the human tribe, Hominini, but there is abundant fossil evidence to indicate that we were preceded for millions of years by other hominins, such as Australopithecus, and that our species also lived for a time contemporaneously with at least one other member of our genus, Homo neanderthalensis (the Neanderthals). In addition, we and our predecessors have always shared the Earth with other apelike primates, from the modern-day gorilla to the long-extinct Dryopithecus. That we and the extinct hominins are somehow related and that we and the apes, both living and extinct, are also somehow related is accepted by anthropologists and biologists everywhere. Yet the exact nature of our evolutionary relationships has been the subject of debate and investigation since the great British naturalist Charles Darwin published his monumental books On the Origin of Species (1859) and The Descent of Man (1871). Darwin never claimed, as some of his Victorian contemporaries insisted he had, that “man was descended from the apes,” and modern scientists would view such a statement as a useless simplification—just as they would dismiss any popular notions that a certain extinct species is the “missing link” between man and the apes. There is theoretically, however, a common ancestor that existed millions of years ago. This ancestral species does not constitute a “missing link” along a lineage but rather a node for divergence into separate lineages. This ancient primate has not been identified and may never be known with certainty, because fossil relationships are unclear even within the human lineage, which is more recent. In fact, the human “family tree” may be better described as a “family bush,” within which it is impossible to connect a full chronological series of species, leading to Homo sapiens, that experts can agree upon.
The primary resource for detailing the path of human evolution will always be fossil specimens. Certainly, the trove of fossils from Africa and Eurasia indicates that, unlike today, more than one species of our family has lived at the same time for most of human history. The nature of specific fossil specimens and species can be accurately described, as can the location where they were found and the period of time when they lived; but questions of how species lived and why they might have either died out or evolved into other species can only be addressed by formulating scenarios, albeit scientifically informed ones. These scenarios are based on contextual information gleaned from localities where the fossils were collected. In devising such scenarios and filling in the human family bush, researchers must consult a large and diverse array of fossils, and they must also employ refined excavation methods and records, geochemical dating techniques, and data from other specialized fields such as genetics, ecology and paleoecology, and ethology (animal behaviour)—in short, all the tools of the multidisciplinary science of paleoanthropology.
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heredity (genetics): Human evolution
Many of the techniques of evolutionary genetics can be applied to the evolution of humans. Charles Darwin created a large controversy in Victorian England by suggesting in his book The Descent of Man that humans and apes share a common ancestor. Darwin’s assertion was based on the many shared anatomical features of apes and humans. DNA analysis has supported this hypothesis. At the...
This article is a discussion of the broad career of the human tribe from its probable beginnings millions of years ago in the Miocene Epoch to the development of tool-based and symbolically structured modern human culture only tens of thousands of years ago, during the geologically recent Pleistocene Epoch. Particular attention is paid to the fossil evidence for this history and to the principal models of evolution that have gained the most credence in the scientific community. See the article evolution for a full explanation of evolutionary theory, including its main proponents both before and after Darwin, its arousal of both resistance and acceptance in society, and the scientific tools used to investigate the theory and prove its validity.
Background and beginnings in the Miocene
It is generally agreed that the taproot of the human family shrub is to be found among apelike species of the Middle Miocene Epoch (16.4 to 11.2 million years ago [mya]) or Late Miocene Epoch (11.2 to 5.3 mya). Genetic data based on molecular clock estimates support a Late Miocene ancestry. Various Eurasian and African Miocene primates have been advocated as possible ancestors to the early hominins, which came on the scene during the Pliocene Epoch (5.3 to 2.6 mya). Though there is no consensus among experts, the primates suggested include Kenyapithecus, Griphopithecus, Dryopithecus, Graecopithecus (Ouranopithecus), Samburupithecus, Sahelanthropus, and Orrorin. Kenyapithecus inhabited Kenya and Griphopithecus lived in central Europe and Turkey from about 16 to 14 mya. Dryopithecus is best known from western and central Europe, where it lived from 13 to possibly 8 mya. Graecopithecus lived in northern and southern Greece about 9 mya, at roughly the same time as Samburupithecus in northern Kenya. Sahelanthropus inhabited Chad between 7 and 6 million years ago. Orrorin was from central Kenya 6 mya. Among these, the most likely ancestor of great apes and humans may be either Kenyapithecus or Griphopithecus.
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Among evolutionary models that stress the Eurasian species, some consider Graecopithecus to be ancestral only to the human lineage, containing Australopithecus, Paranthropus, and Homo, whereas others entertain the possibility that Graecopithecus is close to the great-ape ancestry of Pan (chimpanzees and bonobos) and Gorilla as well. In the former model, Dryopithecus is ancestral to Pan and Gorilla. On the other hand, others would have Dryopithecus ancestral to Pan and Australopithecus on the way to Homo, with Graecopithecus ancestral to Gorilla. This morphology-based model mirrors results of some molecular studies, which show chimpanzees, bonobos, and humans to be more closely related to one another than any of them is to gorillas; orangutans are more distantly related.
In a phylogenetic model that emphasizes African Miocene species, Samburupithecus is ancestral to Australopithecus, Paranthropus, and Orrorin, and Orrorin begets Australopithecus afarensis, which is ancestral to Homo.
The Miocene Epoch was characterized by major global climatic changes that led to more seasonal conditions with increasingly colder winters north of the Equator. By the Late Miocene, in many regions inhabited by apelike primates, evergreen broad-leaved forests were replaced by open woodlands, shrublands, grasslands, and mosaic habitats, sometimes with denser-canopied forests bordering lakes, rivers, and streams. Such diverse environments stimulated novel adaptations involving locomotion in many types of animals, including primates. In addition, there were a larger variety and greater numbers of antelope, pigs, monkeys, giraffes, elephants, and other animals for adventurous hominins to scavenge and perhaps kill. But large cats, dogs, and hyenas also flourished in the new environments; they not only would provide meat for scavenging hominins but also would compete with and probably prey upon them. In any case, our ancestors were not strictly or even heavily carnivorous. Instead, a diet that relied on tough, abrasive vegetation, including seeds, stems, nuts, fruits, leaves, and tubers, is suggested by primate remains bearing large premolar and molar teeth with thick enamel.
Behaviour and morphology associated with locomotion also responded to the shift from arboreal to terrestrial life. The development of bipedalism enabled hominins to establish new niches in forests, closed woodlands, open woodlands, and even more open areas over a span of at least 4.5 million years. Indeed, obligate terrestrial bipedalism (that is, the ability and necessity of walking only on the lower limbs) is the defining trait required for classification in the human tribe, Hominini.
Striding through the Pliocene
The anatomy of bipedalism
Bipedalism is not unique to humans, though our particular form of it is. Whereas most other mammalian bipeds hop or waddle, we stride. Homo sapiens is the only mammal that is adapted exclusively to bipedal striding. Unlike most other mammalian orders, the primates have hind-limb-dominated locomotion. Accordingly, human bipedalism is a natural development from the basic arboreal primate body plan, in which the hind limbs are used to move about and sitting upright is common during feeding and rest.
The initial changes toward an upright posture were probably related more to standing, reaching, and squatting than to extended periods of walking and running. Human beings stand with fully extended hip and knee joints, such that the thighbones are aligned with their respective leg bones to form continuous vertical columns. To walk, one simply tilts forward slightly and then keeps up with the displaced centre of mass, which is located within the pelvis. The large muscle masses of the human lower limbs power our locomotion and enable a person to rise from squatting and sitting postures. Body mass is transferred through the pelvis, thighs, and legs to the heels, balls of the feet, and toes. Remarkably little muscular effort is expended to stand in place. Indeed, our large buttock, anterior thigh, and calf muscles are virtually unused when we stand still. Instead of muscular contraction, the human bipedal stance depends more on the way in which joints are constructed and on strategically located ligaments that hold the joints in position. Fortunately for paleoanthropologists, some bones show dramatic signs of how a given hominin carried itself, and the adaptation to obligate terrestrial bipedalism led to notable anatomic differences between hominins and great apes. These differences are readily identified in fossils, particularly those of the pelvis and lower limbs.
Although we are bipedal, our pelvis is oriented like that of quadrupedal primates. The early bipedal hominins assumed erect trunk posture by bending the spine upward, particularly in the lower back (lumbar region). In order to transfer full upper-body mass to the lower limbs and to reposition muscles so that one could walk without assistance from the upper limbs and without wobbling from side to side, changes were required in the pelvis—particularly in the ilia (the large, blade-shaped bones on either side), the ischia (protuberances on which body rests when sitting), and the sacrum (a wedge-shaped bone formed by the fusing of vertebrae). Hominin hip bones have short ilia with large areas that articulate with a short, broad sacrum. Conversely, great-ape hip bones have long ilia with small sacral articular areas, and sacra of the great apes are long and narrow. The human pelvis is unique among primates in having the ilia curved forward so that the inner surfaces face one another instead of being aligned sideways, as in apes and other quadrupeds. Curved ilia situate some of the gluteal muscles on the side of the hip joint, where they steady the pelvis as the foot swings forward during a step. This special mechanism allows us to walk smoothly, with only slight oscillations of the pelvis and without gross side-to-side motions of the upper body. Humans have short ischia (and long lower limbs), facilitating speedy actions of the hamstring muscles, which extend the thigh at the hip joint, while great apes have long ischia (and short hind limbs), which give them powerful hip extension for climbing up trees. Characteristically, a human thighbone is long and has a very large, globular head and a short, round neck; at the knee a prominent lateral ridge buttresses the groove in which the kneecap lies. The femurs are farther apart at the hips than at the knees and slant toward the midline to keep the knees close together. This angle allows anthropologists to diagnose bipedalism even if the fossil is only the knee end of a femur. The femurs of quadrupedal great apes, on the other hand, do not converge toward the knees, and the femoral shafts lack telltale angling.
Human feet are distinct from those of apes and monkeys. This is not surprising, since in humans the feet must support and propel the entire body on their own instead of sharing the load with the forelimbs. In humans the heel is very robust, and the great toe is permanently aligned with the four diminutive lateral toes. Unlike other primate feet, which have a mobile midfoot, the human foot possesses (if not requires) a stable arch to give it strength. Accordingly, human footprints are unique and are readily distinguished from those of other animals. | <urn:uuid:76f9a911-8795-420f-9685-8eed1e4a8f41> | {
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Fourth grade content connection sample: fourth graders are extending the learning of similarities and differences through the analysis of critical findings and the development of analogies in social studies, compare and contrast is an essential skill for understanding ideas. Although these strategies can benefit all young learners, the compare-contrast students have the opportunity to make an immediate and concrete connection between what they are learning and themselves a third compare the impact of text structure instruction and social. Compare and contrast social learning theory and cognitive behavioral theory abstract learning theories play an important role in our life the social learning theory and cognitive behavioral theories has an significant impact on our life. Teaching strategies: authentically compare & contrast by: jacqui murray to students, knowing how to compare and contrast sounds academic, not real-world, but we teachers know most of life is choosing between options social studies view lesson plan most recent popular. The secondary social studies course of study in newport news public schools is designed to develop the knowledge and skills of extended learning youth development our schools directory of what tools can social scientists use to compare and contrast people, places, ideas, and events. Third grade lesson plans for english and language arts subjects compare and contrast lesson plan materials required: title - hunting whales lesson #4 by - debbie haren primary subject - social studies secondary subjects - language arts grade level. The concept of a core or integrated curriculum has been evolving throughout there are several learning theories that support an interdisciplinary approach with the focus on connecting science, social studies and language. Compare and contrast learning theories education essay print reference this a good side by side comparison of the differences of classical and operant conditioning would be in the study the social learning theory is based on observing whereas classical conditioning is based on the.
Long ago & now unit helps students understand how their lives relate to things and events long ago includes: compare and contrast artifacts long ago & today family tree grandparent (or older relative) interview timeline. Compare and contrast the pros and cons of the following: (1) integrated social studies learning gurublue saturday, january 16, 2010 at 7:22pm these sites will give you some information. Integrated curriculum in the primary program literature, drama, social studies, math, science, health, physical education, music, and visual arts possibilities for integrated learning and teaching planning for an integrated curriculum. Students will view a video clip about brazil's economy and compare it to america's (technology, social studies) project based learning for social studies project based learning: integrated curriculum: definition, benefits & examples related study materials related recently updated. Changing weather, changing seasons a first grade unit history/social studies students compare and contrast the absolute and relative locations of people and places and describe the k-22 use a variety of media and technology resources for directed and independent learning activities.
The indiana's k - 8 academic standards for social studies are organized around four content areas the content area standards and the types of learning experiences they provide to students in grade 7 are described below on the compare, and contrast the historical origins, central. Running head comparison matrix comparison matrix and essay venus horta gcu january 13, 2013 eed465 comparison matrix and essay part one social studies.
Comparison matrix no description the ultimate goal is for students to effectively learn and apply social studies social studies learning comparison thank you diana beltran there are various ways to implement integrated social studies learning and its great because it is flexible. This study considers the enrichment of social studies methods through the integration of videoconferencing in a telecollaborative format to determine if telecollaboration could be successfully and seamlessly integrated within the course. Traditional classroom teaching: petra university dr ahmad al-hassan petra university abstract the purpose of this study was to compare and contrast (a) the effectiveness of e- studies comparing the learning effects of online versus face-to-face instruction.
This education articles offers easy ways to integrate science across the curriculum the arts, social studies and health k-12 news, lessons & shared resources by teachers, for t-charts or other graphic organizers to compare and contrast the main ideas.
A major driving force behind integrated teaching and learning is the curriculum have placed a greater amount of emphasis on the fact that student experience is essential for meaningful learning to occur integrated curriculum seems to be using the social studies/language arts. Lesson plan 1: social studies pages: 21-26 lesson plan 2: reading active learning also benefits students at this age i want my students to be able compare and contrast different native american tribal. Integrative learning and interdisciplinary studies social studies and whole language the ability to compare and contrast them to reveal patterns and connections. International journal of humanities and social science in contrast, constructivist or numerous studies have been completed to compare students' learning in traditional and constructivist classrooms these studies generally based their conclusions on test or. | <urn:uuid:eeca4acd-80e6-4069-a891-03e55ea2fc36> | {
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Each year the United States of America and her territories celebrate Independence Day with fireworks, patriotic displays, barbecues, picnics and parties. Here in the U.S. Virgin Islands, we get to commemorate not one, but two historical acts of freedom back to back: Emancipation Day and Independence Day. Both Emancipation Day and Independence Day are significant for the freedoms they provided, but took place almost three-quarters of a century apart, and provided different kinds of emancipation to people in different circumstances. Here on St. Croix, we like to take these two days to celebrate the cultural heritage of the island, spend time with family and friends, and enjoy some fireworks of our own.
Most of you reading this know that Independence Day is a federal holiday celebrated by citizens of the USA on the fourth of July. Independence Day commemorates the adoption of the Declaration of Independence on July 4, 1776, in which the Second Continental Congress declared that the thirteen American colonies were now a new nation of their own, the United States of America, and no longer part of the British Empire. The Declaration also established the rights of mankind and the grievances the colonists had against British rule. As a result, Independence Day marks not only the formation of the United States of America, but all of the freedoms it has come to represent for its citizens.
On the other hand, many of you may not yet be familiar with Emancipation Day. Here in the US Virgin Islands, Emancipation Day is a public holiday celebrated on the third of July to commemorate the abolition of slavery in the Danish West Indies (now the US Virgin Islands). The Danish West India Company settled on what is now known as the US Virgin Islands in the 17th century, and brought the trans-Atlantic slave trade to these islands in 1673. Slaves, mainly working on the sugarcane plantations, were forced to work in harsh conditions and treated inhumanely, which lead to several large slave revolts.
Peter von Scholten became governor of the islands in 1835 and attempted to ease the burden of the slaves. On September 18th, 1847, Governor General Peter Von Scholten made a public announcement regarding the gradual end to slavery and had a Royal Decree was read at all churches on the islands. However, many refused to wait 12 years for freedom so freed slave and skilled craftsman Moses Gottlieb, (known as ‘General Buddhoe’) led a non-violent slave revolt on the island of St. Croix in 1848 and demanded immediate Emancipation. The revolt led Von Scholten to emancipate all slaves immediately, 10 months before the ‘scheduled emancipation’.
Slavery on the Danish West Indian Islands was officially abolished on July 3, 1848, seventeen years before slavery would be abolished in the USA. Now, the US Virgin Islands is honored as the ‘birthplace of emancipation in the USA.’ The anniversary of this event was declared a territorial public holiday in the US Virgin Islands, and it’s commemoration is immediately followed by the celebration of Independence Day on July 4th.
These two holidays give us many wonderful freedoms to celebrate here in the USVI. The largest celebration on St. Croix takes place out in Frederiksted, known locally as ‘Freedom City’, the site of the 1848 slave revolt that finally brought about the end of slavery in the USVI. Typically held on July 4th, the day’s activities include old-fashioned games for children, cultural entertainment, orators and historians, musical entertainment, quadrille dancers, and food, drink and craft vendors. Once the sun has set, fireworks light up the night sky from the Frederiksted Pier and can be seen for miles around.
Join in and help us commemorate Emancipation Day and celebrate Independence Day with family, friends, food and fun! And while you are enjoying all of the activities and entertainment the day brings, remember that we are celebrating the freedoms fought for by our ancestors, whether here on St. Croix or back in the thirteen colonies of America. Here’s to freedom, fun, and fireworks!
– Jennie Ogden, Editor | <urn:uuid:1f4743d6-0eac-434a-9d91-4867b024289d> | {
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# Co-variance: An intuitive explanation!
A comprehensive but simple guide which focus more on the idea behind the formula rather than the math itself — start building the block with expectation, mean, variance to finally understand the large picture i.e. co-variance
co-variance calculation in all its glory!
#### Introduction
Contrary to the popular belief, a formula is much more than just mathematical notations. It tries to express an idea, which get hidden under the math and is not evident unless you really look for it. The main problem with this kind of representation (as it usually happens with me), is that after sometime you tend to forget the formula. So, here is my attempt to explain one topic such that it sticks with the audience. Before diving right into it, I will try to explain some prerequisite topics. If you are already familiar with them, feel free to skip. If not, ride along :)
#### Expectation and Mean
Let’s start with a relatively easier topic which is the one of the basic blocks required to understand co-variance. In probability theory, expectation represents “the expected value of a **discrete random variable*, which is the *probability-weighted average* of all its possible values” *and it is formalized as,
the expectation over all values of ‘X’ and their probabilities ‘p_i’
Here ‘X’ is the variable which can take several shapes — ‘x_i’, where each have it’s own probability ‘p_i’ of occurrence. Notice expected value is a single number representation of all the values a variable can take considering their probabilities. One special case to remember is when all ‘p_i’ are equal i.e. probability of occurrence of all values are equal. In this case, expected value transforms into mean or average. To give an example, suppose a variable which simulates the rolling of an unbiased dice, so the possible values it can take can be 1 to 6. Also the probability of occurrence of any of these numbers will be equal. Coming back to the generalization, the transformation of expectation to mean is showcased below,
Expectation to mean — how making probability constant makes this possible.
Notice as equal weights are given to all of the values of the variable, the mean is proportional to the values itself, hence it tends to incline towards denser concentration of points. See below a simulation of distribution of points (blue dots) and how change in their position leads to change in the mean (red dot) itself.
Changing points and chasing mean!
Also be vary of practical simulations, as most of the time they differ from the theoretical simulations. Consider the dice roll example, where we very easily stated that they have equal probability, but a programmed simulation may show some variations. Below, I simulated 10,000 rolls of an unbiased dice. Look at the occurrence distribution of the dice faces.
Simulating an unbiased dice roll 10,000 times!
Now compare the theoretical and practical calculation of mean notice there is a difference, even though small, but in practical scenario this will do.
Comparing theoretical and practical mean calculation#### Variance
Observe the following plots, can you find anything common in them?
Changing points and … wait a sec!?
The answer is — all of them have the same mean! But they look so different, right? And what is so different in all of the them? It seems that they have different ‘spread’ or ‘width’. Variance is basically the measure of this spread or width of the data. In statistics, “variance is the **expectation* of the squared *deviation* of a *random variable* from its *mean*.” *Let’s try to fit this definition to our understanding of expectation,
Variance — definition to formula
And just like that we have the formula of variance! Notice first we compute the mean of all the values of ‘X’. Then we find the numerator, which is square of the difference of each value with this mean. The square part is required as we don’t care about the direction of spread, hence we don’t want the spread in opposite directions i.e. with different polarity, to cancel out each other. Some may say, if we square to find the numerator, why not later take a square root? And this idea is exactly represented by standard deviation. So in other words, variance is the square of the standard deviation. With this in mind, let’s look at the same plots as before (now separated and static), but now with variance and standard deviation computed.
Changing points, static mean but changing variance and standard deviation!
Now we are ready for the main topic but before that there is one more interesting derivation of variance. This isn’t required to understand co-variance but curious readers may want to see it anyways. It represents an idea that, *“variance of a variable is expectation of the squared variable minus the square of the expectation itself”. *It is derived below,
Just another interesting derivation of variance!
While we are at it, let’s compute the variance of the dice roll simulation from before. Also let’s compute the same variance in three different ways, one representing lazy python way and the remaining two representing the formulas we discussed.
Computing variance in python — sorry for my pythonic one liner codes :)
Note that the variance is same to a certain decimal value, the small difference there is due to the floating point errors. Also, in Python prefer the way 1, I have coded way 2 and way 3, just to showcase the formulas we discussed.
#### Co-variance
Till now we have been looking at only one variable at a time i.e. our data was 1D or 1-dimensional. Co-variance is defined for higher dimensional data. So as the name suggests, instead of just one variable, it considers multiple (exactly 2) variables and compute variance. Before going further, let’s discuss the data first. When I say 2D, I mean each instance of data is represented by two numbers. And in basic planar geometry, we know two numbers can be associated with a point, hence each instance is represented by a single point. A sample data with 10 instances and their visualization on a 2D plane is shown below,
Now back to it, more formally co-variance is “a measure of the joint variability of two **random variables*”. *The idea is if both variables follow same increasing behavior we have high co-variance. By this I mean, if one variable increases so does the other. In all of the other cases, like where one increases (or decreases) and the other decreases (or increases), we will have negative co-variance. As its a extension of variance, we can express co-variance formula from variance formula as follows,
Variance to co-variance
Please note the subtle second line, which says that co-variance of same variable is equal to variance of the variable. And later all we did was to replace the ‘x’ related term from the second portion of numerator with ‘y’, and we have the co-variance formula! Also note that ‘ux’ and ‘uy’ are the mean of variable ‘X’ and ‘Y’ respectively.
One interesting intuition emerges if we look closely at the numerator. But to generalize this, suppose we computed the mean of both variables (‘ux’ and ‘uy’) and took any one point (‘x_i’ and ‘y_i’) from dateset and plotted both of them as points in a 2D plane. Then we can form a rectangle with these two points being at the exact opposite positions. Going in this direction leads to,
co-variance is nothing but average sum of all rectangle areas
So, it’s basically representing the area of a rectangle which is plotted between the mean and a data point. So each point of the dataset will make a rectangle with mean point. But as we draw rectangles for the complete dataset and find their area using the above formulae, we observe some rectangles have negative area! Nothing to worry, as it just showcase that this data point (for which we get -ve area) has different behavior for its variables, i.e. one is high but the other is low, which goes against the idea of co-variance. Now as we take the summation of all the data points’s rectangle area, what we are doing is adding the +ve areas and subtracting the -ve areas. And finally the resulting value after averaging represents the magnitude of co-variance. One examples on a sample dataset is shown below, where we showcase different points of dataset and the rectangles formed with their areas,
Rectangles formed by the mean-point with rest of the points.
Negative area is represented as red, positive area by green.Let’s also look at the different datasets and the rectangles formed. Also notice that the Figure 1 represents the case with largest area and hence largest co-variance (we discussed how they are proportional). As stated before, the behavior shown here is the ideal one i.e. as one variable increases, so does the other. And as points starts to deviate from this straight line behavior, as shown in the subsequent figures, the amount of red rectangles increases and hence the magnitude of co-variances decreases.
A decrease in area from figure 1 to figure 4, due to change in expectation of “both variables should show similar behavior”
#### Conclusion
When we represent a formula in an easily interpretable form such as a diagram or plot, it becomes much easier to understand and also easier to grasp hidden insights. For example if we represent co-variance in form of rectangles and their areas, we can quickly answer questions like which will have higher co-variance, an exponential decay or growth plot? Or what will happen if there are many +ve points near mean but one -ve point far away from the mean? (read, outliers). Try to think of these questions in the form of rectangles and areas and the answer will come out quickly. I hope it does :)
#### Reference
For any question, feel free to connect with me on linkedin or visit more articles like this on my website.
Cheers. | crawl-data/CC-MAIN-2024-18/segments/1712296817222.1/warc/CC-MAIN-20240418160034-20240418190034-00709.warc.gz | null |
“Non-renewable” energy sources, as well as “renewable” energy and “alternative fuels” help to satisfy the nation’s energy needs. Coal, a non-renewable fossil fuel, plays a large role in the generation of electricity as well as in industrial processes such as the manufacturing of steel. Nuclear, hydro, solar, wind, biomass, and geothermal are all considered “renewable” forms of energy and comprise varying levels of supply in this country. They are classified as renewables since their source is seen as being virtually unlimited. Of these, solar, wind, biomass, biodiesel, and geothermal are all considered “alternative” energy sources since they are not the “traditional” kind (fossil fuels, nuclear and, hydro).
Figure 1 below shows the break-out of fuel sources used in the generation of electricity. As you can see, the single largest fuel is coal, although this is changing as historically low natural gas prices are causing some “fuel switching.” This is followed by natural gas, nuclear, hydro, and “other.” This final category is comprised of energy sources such as fuel oil (a crude distillate), wind, solar, biomass, and geothermal. Note the very small percentage currently represented by all of these combined. It will literally take decades for alternative fuels to make a substantial contribution to the energy portfolio in the United States. Thus, there is a need to continue to use fossil fuels and nuclear power to “bridge” the gap. How the former are delivered to market and how they are priced is the main focus of this course.
The following chart illustrates the various types of energy in the US and the corresponding consumption types. Again, note that the current contribution of renewable energy sources is very small.
Now that we have clarified the difference between renewable and non-renewable sources of energy, let’s have a look at the production and consumption of energy in the United States on a macro level. | <urn:uuid:8e990270-04c9-43c2-8232-29a8bf88198f> | {
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Human activities are the main factors triggering biodiversity loss in West Asia. Habitat destruction and fragmentation due to urban expansion, tourism developments, dredging and reclamation of coastal areas are serious problems in the region, especially along the coasts.
Multilateral agreements to minimize these threats are gaining ground in the region. In 2004, Lebanon, Oman and United Arab Emirates joined the International Treaty on Plant Genetic Resources for Food and Agriculture, which entered into force on 29 June 2004. Jordan, Kuwait and Syria had joined previously (FAO 2004).
Jordan and Syria also became parties to the Cartagena Protocol on Biosafety in 2004 (SCBD 2004a). The West Asian countries face challenges in complying with the provisions of the protocol. For example, they lack expertise in the safe transfer and handling of genetically modified organisms and their products (SCBD 2004b). The capacity of national institutions in the region must be strengthened and national biosafety frameworks developed. Efforts must be made to avoid contamination of local crop varieties and wild relatives with genetically modified strains.
Box 3: Putting agrobiodiversity into practice
The UNDP/Global Environment Facility project on the Conservation and Sustainable Use of Dryland Agrobiodiversity in West Asia aims to halt and reverse the loss of biodiversity in ten major crops and their wild relatives (Valkoun and others 2004, UNDP 2003b).
For example 'Hourani', a wheat variety planted in Syria and Jordan for 1 000 years, nearly became extinct, replaced by highly productive Italian and Mexican wheat varieties in the 1970s. Now, genetic erosion has been slowed by promoting the reintroduction of local wheat varieties into farming systems (Charkasi 2000). Other species targeted by the project have been incorporated in reforestation programmes (ICARDA 2002 and 2004).
Poverty alleviation through income-generating micro-projects is another significant result. Bee keeping and production of organic products are becoming popular on project pilot sites. | <urn:uuid:8f123e50-e0ad-4ce7-b519-7076df2897f7> | {
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Half of the mammals, birds, amphibians and reptiles living in Colorado’s mountains are at risk of becoming extinct over the next century, according to a recent paper co-authored by a University of Colorado professor.
Climate change predictions that calculate only for temperature changes estimate extinction risks of about 5 percent of species. In this study, which looks at temperature increase and precipitation changes for 16,848 vertebrate species on 156 mountains, the possible local extinction rate increases 10-fold to roughly 50 percent over the next 100 years.
“Everyone thinks about just temperature change, but the truth is that for vertebrates and other mountain organisms, what these models are showing is that precipitation change can be so much more severe,” says Christy McCain, assistant professor in the University of Colorado’s Department of Ecology and Evolutionary Biology and curator of vertebrates for the CU Museum of Natural History. McCain co-authored the paper with University of Connecticut professor Robert Colwell.
Most climate change models don’t predict what the effect will be on precipitation, a more expensive and more tricky variable to calculate.
“So we said, let’s run the models for wetter, drier and average and see how species might respond,” McCain says. “How much of their niche that they have now would be there in 100 years under all three of those scenarios?” The expectation is that animals will move up in elevations to stay in cooler temperatures. Only a few of them, a few specialist species that live at the tops of peaks, would need to essentially float off the mountain to stay in cool enough temperatures.
In temperate climates, higher elevations tend to be wetter, meaning desert species could find themselves tracking their ideal temperatures into much wetter climates, McCain says. And at higher elevations, much of the precipitation falls in the form of snowpack, when many of the native species are dormant and unable to access it.
“We were just trying in our models to say, OK, there is uncertainty around how precipitation is going to change, but if we look at all that uncertainty, what is the risk?” McCain says. They ran their models assuming various levels of adaptability to wetter or drier climates for species. “Regardless of which model you use … the risks are so much higher because of this disconnect between tracking a cooler temperature and moving outside your precipitation that you’re used to having.”
In Central America, where drastically drier conditions are predicted for the next century, amphibians like salamanders and frogs face a local extinction risk of up to 91 percent and 71 percent, respectively. In the Rockies, even common species like certain chipmunks and shrews are at risk, as is the more isolated American pika. North American local extinction risks go as high as 49 percent of vertebrate species per mountain range.
Research from a CU-Boulder study team on pikas in the southern Rocky Mountains has shown pika populations abandoning drier sites.
“We suspect that a lack of snowpack leads to a lack of insulation for pikas in the winter. … If they’re exposed to cold temperatures, because there isn’t sufficient snowpack, they could potentially freeze to death,” says Liesl Erb, the doctoral student who led the study team that assessed historic sites for pikas. “It’s also possible that the lack of precipitation could lead to lack of sufficient water in the vegetation that they eat. At this point it could be a combination of those factors.”
John Williams, a professor at the University of Wisconsin, has also studied climate change models that factored for both precipitation and temperature.
“What we found was that these climate model projections for the 21st century, if you compare those projections to the late 20th climates, you see areas where there’s novel climates emerging and current climates disappearing,” he says. “Meaning that in some areas of the world, climates that exist today will disappear or greatly shrink in size by the end of the century. In other areas of the world, mainly in the lower tropics, there will be expansions or appearances of novel climates that have not been seen within the earth’s system over the last several million years.”
These new climates might expand the habitat for some species, like those that live in relatively warm environments, but it could also eliminate the environments of those that prefer relatively cold environments, Williams says.
“There is some question of surprise,” he says. “What will happen in these novel climates that are outside of our experience?” | <urn:uuid:1d48c9cd-3a3f-4ca4-9e3d-0831c8a95d58> | {
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— OR —
Using this CVC Word Sounds Worksheet, studentschoose the CVC word that matches each pictures to build their reading fluency.
CVC words make up a lot of the words your students encounter. This worksheet gives them practice sounding out and spelling CVC words.
Look at the pictures. Say the words. Think about beginning, middle, and ending sounds. Choose the word that matches each pictures. Trace the correct word.
If you are using this worksheet, your students are probably learning about CVC words.
Use this CVC Vowel Activity as an additional resource for your students.
Introduce this worksheet by practicing reading CVC words with your students. Students can use their hands or arms to break down the word. Then, students complete the worksheet independently or with a partner. Once finished, challenge students to write sentences. If students are struggling with this worksheet, encourage them to read words aloud and break them down!
Be sure to check out more CVC Activities.
Tell others why you love this resource and how you will use it.
You must be logged in to post a review.
Make Resources FREE with a Membership! | <urn:uuid:2ede3a3c-02c1-4fb8-b56e-6f59209c1a25> | {
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In the realm of quantum mechanics, atoms and subatomic particles just don’t follow the rules that we’re governed by in the larger world of classical mechanics. For example, the theory of quantum mechanics predicts that two or more particles can become “entangled” so that even after they are separated in space, when an action is performed on one particle, the other particle responds immediately. Scientists still don’t know how the particles send these instantaneous messages to each other, but somehow, once they are entwined, they retain a fundamental connection [LiveScience].
Now, a new study has dragged entanglement a little bit closer to our classical world. Researchers managed to entangle two pairs of vibrating ions so that when the motion of one pair of ions was changed, the other pair reflected the change as well. Previously, researchers have entangled particles in much more esoteric ways, coordinating the spin of electrons or the polarization of photons. With this study, says coauthor John Jost, “We’ve entangled something that has never been entangled before, and it’s the kind of physical, oscillating system you see in the classical world, just much smaller” [LiveScience].
In the study, published in Nature, the two ion pairs were held about a quarter of a millimeter apart in an ion trap, which is a significant separation for atoms. In each pair, a beryllium ion was partnered with a magnesium ion. “You can think of them like two balls connected by a spring that vibrate back and forth in unison,” says Jost. The first step to achieving these synchronized vibrations relied on standard techniques to entangle the spins of the beryllium ions in each pair. The trick was then to transfer this conventional form of entanglement into the vibration of the ion pairs, using lasers. “Depending on its internal spin state, the beryllium ions will absorb certain frequencies of laser light, which excites them and sets them vibrating,” explains Jost. The two entangled pairs of beryllium and magnesium ions then began to vibrate in lockstep [Nature News].
Christopher Monroe, a quantum physicist who wasn’t involved in the current study, says he finds the work enticing: “We all want to move quantum mechanics to the macroscopic world we live in.” … The separation between the quantum world and the macroscopic world is still unclear and interests many researchers. Now that entanglement has been demonstrated in a mechanical system, says Monroe, scientists may be able to apply the findings to larger and larger mechanical systems. Quantum mechanics shouldn’t care whether a system involves a couple atoms or trillions of atoms, Monroe says. “The quantum physics is exactly the same” [Science News].
80beats: Quantum Teleportation is a Go!
80beats: Harnessing Quantum Weirdness to Make Spy-Proof Email
80beats: Entangled Particles Seem to Communicate Instantly—and Befuddle Scientists
DISCOVER: Next-Level Quantum Spookiness
DISCOVER: Teleportation Gets Real
DISCOVER: Is Quantum Mechanics Controlling Your Thoughts?
Image: John Jost and Jason Amini | <urn:uuid:dbe51948-98b5-46ec-8a92-e8c863c82d59> | {
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Solve Linear Equations with Rational Numbers
Solve Linear Equations
(with Rational Numbers)
When solving linear equations, the goal is to isolate the unknown (variable) to find its value. This is accomplished by adding, subtracting, multiplying, or dividing. Let’s start with a simple example:
x + 4 = 9
The main rule to keep in mind when trying to isolate a variable is that you always have to keep the equation balanced. This means that whatever you do to one side of the equation, you must also do to the other side. In the equation above, we need to get rid of the "+4" on the left side of the equation. We do this by using the opposite operation, which is subtraction. So we subtract 4, which cancels out the "+4." The x is now isolated, but we must also subtract 4 from the right side:
x + 4 = 9
- 4 - 4
x = 5
For a more complicated equation, there might be more steps. Let us take the example below:
3x - 9 = x + 3
When you have multiple operations involved in a problem, you must always do addition and subtraction first. Multiplication and division come second. You have a few options for what you do first with this problem. Let us combine the x terms first. We can subtract x from each side to get:
3x - 9 = x + 3
- x - x
2x - 9 = 3
Then we will need to add 9 to each side:
2x - 9 = 3
+ 9 + 9
2x = 12
Finally we need to isolate x. 2x is essentially 2 times x. To cancel this out, we use the opposite operation, which is division. We do this by dividing by 2.
2x = 12
2 2
x = 6
You can always check your solution be plugging it back into the original equation.
3x + 9 = x + 3
3(6) + 9 = 6 + 3
18 + 9 = 9
9 = 9
Example 1:
First add 6 to each side to get
Then you need to get rid of the fractions by multiplying each side by 3.
x = 2x + 30
-2x -2x
-x = 30
To make the x positive, we multiply each side by -1. So the answer becomes
x = -30
Example 2:
7y + 5 - y + 1 = 2y - 6
This equation has like terms on one side of the equation. You should combine them before beginning to isolate the variable. So the equation becomes
6y + 6 = 2y - 6
- 6 - 6
6y = 2y - 12
Then subtract 2y from each side and then divide both sides by 4:
4y = -12
4 4
y = -3
Practice Problems:
1.)
2.)
3.)
4.)
1.) x = 4
2.) y = 5
3.) x = 6
4.) x = 10 | crawl-data/CC-MAIN-2020-40/segments/1600400198868.29/warc/CC-MAIN-20200920223634-20200921013634-00395.warc.gz | null |
# Limit $\dfrac{\sin x}{x}$ is $1$ when $x$ tends to zero
## Formula
$$\lim_{x \to 0} \dfrac{\sin x}{x} = 1$$
### Proof
$x$ is a literal and represents an angle of the right angled triangle and $\sin x$ is the sine function. The ratio of $\sin x$ to $x$ is expressed as $\dfrac{\sin x}{x}$. The value of ratio of $\sin x$ to $x$ as $x$ approaches zero is expressed in mathematical form in limit form.
$$\lim_{x \to 0} \dfrac{\sin x}{x}$$
The range of the $\sin x$ function is $[-1, 1]$. It is evident that the values of $\sin x$ function lies from $-1$ to $1$. A special property of the sine function is revealed when you study its functionality closely for angles which are very close to zero. In other words, the values of $\sin x$ function is approximately equals to angles when the angles tend to zero.
#### Example
$(1) \,\,\,\,\,$ $x = 0.176598 \implies \sin 0.176598$ $=$ $0.1756815076\cdots$ $\approx$ $0.176598$
$(2) \,\,\,\,\,$ $x = 0.053874 \implies \sin 0.053874$ $=$ $0.0538479431\cdots$ $\approx$ $0.053874$
$(3) \,\,\,\,\,$ $x = 0.001234 \implies \sin 0.001234$ $=$ $0.0012339996\cdots$ $\approx$ $0.001234$
$(4) \,\,\,\,\,$ $x = 0.000235 \implies \sin 0.000235$ $=$ $0.0002349999\cdots$ $\approx$ $0.000235$
$(5) \,\,\,\,\,$ $x = 0.000056 \implies \sin 0.000056$ $=$ $0.0000559999\cdots$ $\approx$ $0.000056$
The examples clear that the value of $\sin x$ is approximately equal to the angle. So, it is expressed as $\sin x \approx x$.
$$\implies \lim_{x \to 0} \dfrac{\sin x}{x} = \lim_{x \to 0} \dfrac{x}{x}$$
$$\require{cancel} \implies \lim_{x \to 0} \dfrac{\sin x}{x} = \lim_{x \to 0} \dfrac{\cancel{x}}{\cancel{x}}$$
$$\implies \lim_{x \to 0} \dfrac{\sin x}{x} = \lim_{x \to 0} 1$$
$$\therefore \,\,\,\,\, \lim_{x \to 0} \dfrac{\sin x}{x} = 1$$
Therefore, the identity is evident that the value of ratio of $\sin x$ to $x$ is one when limit $x$ tends to zero.
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The crust is the outer hard layer of the planet. The crust is a part of the Geology/Content/Lithosphere. On Earth, the crust is under the troposphere and above the ocean. It is important to remember that not every geological sphere has a crust. The crust on Earth is being created through a process called Continental Drift, but on the moon it is not being made. The crust has two different parts. One is the continental, and the other is oceanic. The continental crust is thicker, and the oceanic crust is thinner. Thicknesses of the crust can be anywhere from 5-70 km. | <urn:uuid:3f65e325-2137-41e4-9969-7c35f58f518a> | {
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what does it mean "passive voice use"?
These attacks were plotted and enforced by terrorists hate groups against this country.
Active voice is always preferable in writing. In active voice, the subject directly does the action. You can change the passive voice in your sentence to active, by saying:
Terrorist hate groups plot and execute attacks against this country.
|link comment||answered Oct 24 '11 at 18:56 Shaila Fernandes Expert|
The active and passive voice are the two choices you have for most sentences. A simple example is:
I love you. (Active)
You are loved by me. (Passive)
The two sentences mean the same thing, but in English, we generally prefer the Active Voice, because we know who is doing what to whom - "I love you."
The Passive voice turns this around, so that the person or thing doing the action comes after the verb, and the person or thing having the verb done to them comes before the verb - "You are loved by me."
You may have noticed a couple of other things about the passive:
1) it uses more words; and
2) we lose the passion! ("You are loved by me" will not win anyone's heart!)
This loss of passion, energy, life is why people tend to use the passive when writing formal documents, to make them sound a bit more distant and less emotional. That's fine, up to a point, but it's a bad habit to get into, because it also makes writing dull and lifeless - that is, boring and tedious to read.
So, how do you spot a passive?
The passive has two main parts:
1) am, is, are, was, were, be, been or being; (any one or more of these words) and
2) a verb in the past tense (it's actually a past participle)
So, in your example, "were + plotted" and "were + enforced" (the second "were" is omitted because the first one works with both verbs)
A common third (though not actually necessary) element of the passive is the word "by" followed by a noun (person or thing). This person or thing is the one actually doing the action. In your example, it's the "terrorist hate groups" that "plotted and enforced".
To change the passive back into the active, you simply take the person or thing after the word "by" and put it back in front of the verb, and get rid of the "am, is, are..." word. If you are a native speaker, you will now automatically make the necessary changes to the verb. If you speak English as a Second Language, you may find that tricky, and that's a bit too difficult to explain here.
So, in "You are loved by me." you do this:
1) put the "me" back in front of the verb "loved"
2) get rid of the "are";
3) put the "you" after the verb;
4) get rid of "by"; and
5) tidy up
That will leave you with "me loved you" which, when tidied up, becomes "I love you."
|link comment||edited Oct 27 '11 at 08:38 Agreeonpurpose Contributor|
Hero of the day
Person gave the most answers! | <urn:uuid:6182ea7c-96bb-4b06-8e92-49961a8effdd> | {
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# Logical sequence test in logical reasoning
Logical sequence test in logical reasoning involves arranging a given set of elements or events in a logical order based on a specific criterion or rule.
Explanation:
Sequence Identification: To solve these questions, you need to identify the logical order or sequence based on the given information.
Rule Application: Apply the rule or criterion provided to determine the correct sequence.
Logical Deduction: Use logical reasoning to arrange the elements in a way that follows the given rule.
Examples:
1. Example: Arrange the following words based on their alphabetical order:
Apple, Banana, Orange, Mango.
Solution: Apple, Banana, Mango, Orange
1. Example: Arrange the following numbers based on their value: 5, 2, 8, 4.
Solution: 2, 4, 5, 8
Multiple Choice Questions:
1. Question: Arrange the following letters in alphabetical order: F, A, R, T, S.
A) A, F, R, S, T
B) F, A, R, S, T
C) A, F, S, R, T
D) F, A, T, S, R
Answer: A) A, F, R, S, T
2. Question: Arrange the following numbers in ascending order: 9, 3, 6, 1, 5.
A) 1, 3, 5, 6, 9
B) 9, 6, 5, 3, 1
C) 1, 3, 6, 5, 9
D) 9, 5, 6, 3, 1
Answer: A) 1, 3, 5, 6, 9
3. Question: Arrange the following days of the week in their natural order: Monday, Wednesday, Friday, Sunday, Tuesday.
A) Monday, Tuesday, Wednesday, Friday, Sunday
B) Sunday, Tuesday, Wednesday, Friday, Monday
C) Monday, Wednesday, Tuesday, Friday, Sunday
D) Sunday, Monday, Tuesday, Wednesday, Friday
Answer: A) Monday, Tuesday, Wednesday, Friday, Sunday
4. Question: Arrange the following colors in alphabetical order: Blue, Green, Red, Yellow, Orange.
A) Blue, Green, Orange, Red, Yellow
B) Blue, Green, Red, Yellow, Orange
C) Orange, Red, Yellow, Blue, Green
D) Yellow, Orange, Red, Green, Blue
Answer: A) Blue, Green, Orange, Red, Yellow
5. Question: Arrange the following shapes in the order of their sides: Triangle, Circle, Square, Pentagon, Hexagon.
A) Circle, Triangle, Square, Pentagon, Hexagon
B) Circle, Square, Triangle, Pentagon, Hexagon
C) Circle, Square, Pentagon, Triangle, Hexagon
D) Circle, Square, Pentagon, Hexagon, Triangle
Answer: A) Circle, Triangle, Square, Pentagon, Hexagon
6. Question: Arrange the following months in chronological order: June, September, February, December, April.
A) February, April, June, September, December
B) June, September, December, February, April
C) February, June, September, December, April
D) April, June, September, February, December
Answer: C) February, June, September, December, April
7. Question: Arrange the following planets in order of their distance from the Sun: Venus, Mars, Earth, Jupiter, Saturn.
A) Earth, Mars, Venus, Jupiter, Saturn
B) Venus, Earth, Mars, Jupiter, Saturn
C) Earth, Venus, Mars, Jupiter, Saturn
D) Venus, Mars, Earth, Saturn, Jupiter
Answer: B) Venus, Earth, Mars, Jupiter, Saturn
8. Question: Arrange the following musical notes in ascending order: C, E, A, G, D.
A) A, C, D, E, G
B) C, D, E, G, A
C) A, C, E, G, D
D) C, E, G, D, A
Answer: A) A, C, D, E, G
9. Question: Arrange the following animals in alphabetical order: Lion, Elephant, Tiger, Zebra, Giraffe.
A) Elephant, Giraffe, Lion, Tiger, Zebra
B) Elephant, Giraffe, Tiger, Zebra, Lion
C) Elephant, Giraffe, Zebra, Lion, Tiger
D) Elephant, Lion, Tiger, Zebra, Giraffe
Answer: A) Elephant, Giraffe, Lion, Tiger, Zebra
10. Question: Arrange the following countries in order of their population: India, Russia, Brazil, China, United States.
A) Brazil, Russia, United States, India, China
B) Russia, United States, Brazil, India, China
C) United States, China, India, Brazil, Russia
D) China, India, United States, Brazil, Russia
Answer: D) China, India, United States, Brazil, Russia | crawl-data/CC-MAIN-2024-38/segments/1725700651559.58/warc/CC-MAIN-20240914061427-20240914091427-00098.warc.gz | null |
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