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User:TanyaJacob

2011-01-28T09:25:17Z

TanyaJacob:

Hi! This is Tanya. I'm originally from India but I have been living in Singapore for the past 7 years. I'm in first year Sauder :)

'''The Pythagorean Theorem'''

The Pythagorean theorem is a concept in geometry which shows the relation of the sides of a right angle triangle. It can be used to figure out any of the unknown sides of the right angle triangle. This theorem is named after the mathematician Pythagoras. The equation of the theorem is:

c^2= a^2 + b^2

where c is the hypotenuse of the triangle and a and b are the other two sides of the triangle.

This theorem can be proven through similar triangles. By equating the sides of the similar triangles regardless of the size, the Pythagoras theorem can be derived. There are various other proofs for this theorem such as subtraction proof, Algebraic proof etc. Lastly even Pythagoras triples exist which are sizes of the sides which always remain the same due to the fact these numbers are only divisible by 1. For example: 3,4,5.

'''CALULUS IN ECONOMICS'''

Economics is a social science and it's about the distribution, production and consumption of goods and services in an economy. In economics, there is a term called elasticity. Elasticity is the responsiveness of z when there is a slight change in the Y.

Elasticity = (percentage change in Z) / (percentage change in Y)

Using basic calculus, we can figure out the elasticity:

(percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z)

Elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)

Using calculus to calculate the elasticity of demand:

Elasticity of demand is the responsiveness of demand from the consumers when there is a slight change in the price of the goods and services provided to the economy.

Price elasticity of demand: = (dQ / dP)*(P/Q)

Example:

If the demand equation for a product is
Q = 110 - 4P
After differentiation, the answer will be
dQ/dP = -4

Substitute dQ/dP = -4 and Q = 110 - 4P into our price elasticity of demand equation:
Price elasticity of demand: = (dQ / dP)*(P/Q)
= (-4)*(P/(110-4P)
= -4P/(110-4P)

At price=5:

Price elasticity of demand: = -4P/(110-4P)
= -20/90
= -2/9

As the value is less than 1, it is said the good or service is price inelastic. This means any small change in price of a product does not have a huge impact on the demand of the product.

Course:MATH110/Archive/2010-2011/003/Teams/Luzern/Homework 12

2011-01-28T02:48:01Z

TanyaJacob: /* Homework 12 */

=Homework 12=
 

The equation given: 
:<math>P(t) = \frac{1}{1+e^{-t}}</math> 
 

We can set up this equation to be more specific: 
:<math>P(t) = \frac{K}{1+re^{-at}}</math>

To change the height of the horizontal asymptote on the right: 
*Change the value of K 

To change the y-intercept to any number between 0 and K 
*Change the coefficient of <math> e^-t </math> 
*Change the value of r 

BONUS 

To change slope of the curved part, slope can go from very close to zero to almost vertical: 
*Change the coefficient in front of t 
*Change the value of a 

We have an excel graph showing these changes: 

MODEL- Tanya 

Another model which uses the same logistic formula is the growth of number of H1N1 cases in 2009. H1N1 virus is still having an exponential growth but the overall population of the world is still about the same, hence the ratio is getting smaller and smaller. As the population will remain the same, eventually this disease will kill too many people and the growth will taper off. The maximum will be reached at this is called the Stationary phase. 

:<math>P(t) = \frac{K}{1+re^{-at}}</math>

K= the height of the horizontal asymptote which also the carrying capacity 
a= the rate by which the growth is occurring 
1= Initial number of people 
T= Time (days/ months/years) 

Eg if K= 1000 
A= 3.02% 
R=1 
At what value of t will the population affected by H1N1 will reach 95% of the carrying capacity 
:<math>P(t) = \frac{1000}{1+1e^{-0.03t}}</math> 

:<math>{950+950e^{-0.03t}}=1000</math> 
:<math>{950e^{-0.03t}}=50</math> 
:<math>e^{-0.03t}=\frac{1}{19}</math> 
(lne^(-0.03t))=ln(1/19) 
(-0.03t)=ln(1/19) 
t=98.15

Course:MATH110/Archive/2010-2011/003/Groups/Group 01

2010-10-13T16:36:28Z

TanyaJacob:

Group members:
* Catherine Chen
* Curtis Doucette
* Tanya Jacob
* Albert König
* Shauna Maty

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== Deriving to the area of the pentagon using squares ==

[[File:P1120678.JPG]]

[[Media:P1120679.jpg]]

Unable to show steps on wiki. Hand written work will be submitted.

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== Working on Solving Problems ==

1) There is no time difference! It is how the time has been written! One hour consists of 60 minutes. and When we add 20 minutes to this it adds up to 80 minutes bus drive. And this is the same exact amount that the driver needed for returning to the terminal

2) As the question is not saying at what time and at what place the policeman saw the woman, I conclude that the policeman was not there when the lady broke the law. The policeman only "might have" her driving. So she might be driving the right way at that time, but 5minues ago she was breaking the law in the absence of the law.
Another conclusion that can be made from this question is that the question is not including cars or any other types of vehicles that are related to an act of crime while driving. Hence we can also conclude that the woman must have been driving a bicycle instead of a car

3) The probability of labeling Apple and orange box correctly is 100% for people who know what an orange and what an apple looks like. But when we reach box three, it becomes tricky. The reason is that there are two different fruits inside of it and when we choose only one fruit, we will label that box according to the fruit picked. Hence the chance of saying the right name for the last box is 0. Because, if we pick an orange then we label the box as orange-box but in fact it is a orange-apple box. The same procedure happens when we pick apple from that box. The only chance of getting this right is to pick at least 3 different fruits from the third box and when we see that we have picked two different fruits we know that it is a combination.

4) If we look at brother in the first part of the sentence and then the plural form of brothers in the second half, we can easily say that this blind fiddler has only one brother.
Looking at it from another point we know that a fiddler is a person who cheats on people mainly for the sake of “robbing” their money. So we can look at this as a gang where one persons say that everyone in the organization is connected to the blind fiddler but none of us inside the organization are connected to each other. It looks like a pyramid, where the tip can be having multiple lines towards the bottom.

5) From different point of views there different numbers of rotationsa.

a)One way is when the picture on the coin is facing the same direction then it has revolved 2 times. One time at 0 degrees and one time at 180 degrees.

b)If we don’t care about the direction the coin’s picture is looking at we had a 360o rotation about its axis, which means that we had indefinite times of turn, until it reaches its origin.

6. Three kinds of apples are all mixed up in a basket. How many apples must you draw (without looking) from the basket to be sure of getting at least two of one kind?

Probability of taking apple of one of three kinds is 1/3 therefore taking 2 of the same kind is 1/3 X1/3
= 1/9

7. Suppose you have 40 blue socks and 40 brown socks in a drawer. How many socks must you take from the drawer (without looking) to be sure of getting (i) a pair of the same color, and (ii) a pair with different colors?
i) Probability of taking out 1 blue sock for example is 1/2, therefore probability of taking a pair of same color is 1/4
ii) 1/4

8. Reuben says, “Two days ago I was 20 years old. Later next year I will be 23 years old.” Explain how this is possible.
Let's say Reuben's birthday is on Dec 12, two days on Dec 10 he was 20 years old. On Dec 12 he is 21 years old. The next Dec12 he would be 22 years old. Later in Dec 13 the next year he would become 23 years old.

9. A rope ladder hanging over the side of a boat has rungs one foot apart. Ten rungs are showing. If the tide rises five feet, how many rungs will be showing?

5 rungs will be showing as the whole rope is 10 foot with 10 rungs one foot apart

10. Suppose one-half of all people are chocolate eaters and one-half of all people are women. (i) Does it follow that one-fourth of all people are women chocolate eaters? (ii) Does it follow that one-half of all men are chocolate eaters? Explain.

No it doesn't as is does not specify whether all 1/2 of the women are chocolate eaters or all 1/2 of the men are chocolate eaters. The number of chocolate eaters can be distributed between all the men and women to make 1/2.

11. A woman, her older brother, her son, and her daughter are chess players. The worst player’s twin, who is one of the four players, and the best player are of opposite sex. The worst player and the best player have the same age. If this is possible, who is the worst player?
This is not possible. Based on how it is phrased, it has to be either the son or the daughter, because the mother and her OLDER brother are not the same age. Therefore, since the best and worst players are of opposite sex, this cannot be possible.

12. A Manhattan fellow had a girlfriend in the Bronx and a girlfriend in Brooklyn. He decided which girlfriend to visit by arriving randomly at the train station and taking the first of the Bronx or Brooklyn trains that arrived. The trains to Brooklyn and the Bronx each arrived regularly every 10 minutes. Not long after he began his scheme the man's Bronx girlfriend left him because he rarely visited. Give a (logical) explanation.
Because it never specifies what intervals the trains come at, it could be the following: Brooklyn- 11:59, 12:09, 12:19 Bronx- 12:00, 12:10, 12:20. Based on when the man arrives at the train station, he could almost always end up picking Brooklyn because it departs 1 minute early.

13. If a clock takes 5 seconds to strike 5:00 (with 5 equally spaced chimes), how long does it take to strike 10:00 (with 10 equally spaced chimes)?
Although it seems that it would just take 10 seconds for the clock to strike ten, simply double, this cannot be right because between 5 chimes, there is only 4 intervals of time so, letting C=chimes and I=intervals, it can be said that 5C+4I=5 and then the formula for the second one would be 10C+9I=x. So, each interval for the 5 chimes is equal to 5/4. Since there is 9 intervals when the clock strikes 10, you would have 5/4*9 which equals 11.25 therefore, it takes 11.25 seconds for the clock to strike ten.

14. One day in the maternity ward, the name tags for four girl babies became mixed up. (i) In how many different ways could two of the babies be tagged correctly and two of the babies be tagged incorrectly? (ii) In how many different ways could three of the babies be tagged correctly and one baby be tagged incorrectly?
there are 6 ways that two of the four babies can be directly tagged. there is no way that three of the four babies can be directly tagged.

15. Alex says to you, “I'll bet you any amount of money that if I shuffle this deck of cards, there will always be as many red cards in the first half of the deck as there are black cards in the second half of the deck.” Should you accept his bet?
No, you should not accept his bet. No matter how many red cards are in the first half, there has to be the exact same of black cards in the second half as there are red cards in the first half. A half of a deck totals to 26 cards and since there are two colors, red and black, the number of red and black cards will be mirrored oppositely. Example: if Alex splits the deck of cards, and we count what we have in the first half, say 20 black cards and 6 red cards, we know without looking that there are going to be 20 red and 6 black in the other half, simply because there are only 2 colors and 26+26=52

[['''Curtis: 16-20''']]

'''16.''' Suppose that each daughter in your family has the same number of brothers as she has sisters, and each son in your family has twice as many sisters as he has brothers. How many sons and daughters are in the family?

'''First we must translate this information into an equation for the daughters and sons. Let S= sisters and let B= brothers then our equation for the daughters is: S-1=B, and for the sons is: S=2(B-1) Next we solve for B by substituting the information we have that S=B+1: B+1=2(B-1), 1=2B-2-B, 3=2B-B, B=3 therefore by substituting B=3 into S-1=B we get: S-1=3 so S=4. We can then see that there are 4 sisters and 3 brothers.'''

'''17.''' The zero point on a bathroom scale is set incorrectly, but otherwise the scale is accurate. It shows 60 kg when Dan stands on the scale, 50 kg when Sarah stands on the scale, but 105 kg when Dan and Sarah both stand on the scale. Does the scale read too high or too low? Explain.

'''Again, we start with equations. Let D= Dan's weight, let S= Sarah's weight and let x= the amount the scale is off by. Then our equations will be D+x=60, S+x=50, and D+S+x=105. Then we can do some simple algebra and substitution to get D=60-x, S=50-x, and (60-x)+(50-x)+x=105. Finally, we can solve for x: -2x+x=105-60-50, -x=-5, x=5. So, the scale is adding 5 kilograms.'''

'''18.''' Alice takes one-third of the pennies from a large jar. Then Bret takes one-third of the remaining pennies from the jar. Finally, Carla takes one-third of the remaining pennies from the jar, leaving 40 pennies in the jar. How many pennies were in the jar at the start?

'''This time we can simply write an equation for the problem letting x= the number of pennies in the jar: 2(2(2x/3)/3)/3=40 and then by reversing this operation we get: x=3(3(3(40)/2)/2)/2 which is really terrible to look at so it can also be viewed as x=40(3/2)^3 therefore x=135. The number of pennies that was in the jar to begin with is 135.'''

'''19.''' One morning each member of Angela's family drank an eight-ounce cup of coffee and milk, with the (nonzero) amounts of coffee and milk varying from cup to cup. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. What is the least number of people in the family?

'''Once more, you guessed it! We are going to write an equation. Let M= total milk consumed by Angela's family in the morning and C= total coffee consumed by Angela's family in the morning and x= the number of members in Angela's family. Our equation will be (M/4 + C/6)x= M+C. Regrouping, we get 2C(6-x)=3M(x-4). Since both C and M are positive quantities, both (6-x), and (x-4) are also positive, which is only possible when x = 5. Therefore, Anglela's family has 5 members in it.
'''

'''20.''' Of two clocks next to each other, one runs 5 min per hour fast and the other runs 5 min per hour slow. At midnight the clocks show the same time. At what time are they are one hour apart?

'''This one is pretty easy since every hour each clock moves 5 minutes away from the other. ie the gap between them is increased by 10 minutes each hour. 60(minutes in an hour)/10(minutes clocks move apart)=6 so, after 6 hours the clocks will be an hour apart. Therefore, the clocks will be 6 hours apart at 6 am.'''

21. Sven placed exactly in the middle among all runners in a race. Dan was slower than Sven, in 10th place, and Lars was in 16th place. How many runners were in the race?

Sven is the median of the sequence. Dan is the 10th and Lars is the 16th, so there must be at least 16 runners in order to have a 16th placement. Since 16 is an even number the isn't an exact median in the sequence. So 17, the next number would be reasonable. The median would be 9. Sven is placed exactly the 9th, which is the middle among all 17 runners, faster than Dan and Lars.

23. Suppose you overhear the following conversation: Paul: How old are your three children? Paula: The product of their ages is 36 and the sum of their ages is the same as today's date. Paul: That is not enough information. Paula: The oldest child also has red hair. If you were Paul could you determine the ages of Paula's children? Explain.

It is impossible to determine the ages of Paula's children. The first piece of information only gives possible combinations that adds up/ multiplies up to 36. We don't know the date of today, we only know that the sum cannot be larger than 31, and their ages has to be smaller than 10 for each child because their product cannot exceed 36.
The second piece of information
24. Two candles of equal length were lit at the same time. One candle took 6 hr to burn out and the other candle took 3 hr to burn out. After how much time was one candle exactly twice as long as the other candle?

Let the length of the candle be 12cm. For the candle that takes 6 hours to burn out, we call it (a), for the other that takes 4 hours to burn out, we call it (b). With the length of 12 cm, we can calculate the rate of burning. For (a), the rate is 2cm/hr, for (b), the rate is 4cm/hr.

After an hour, (a) would be 10cm while (b) would be 8cm. After two hours, (a) would then be 8cm while (b) would be 4cm. This is when the length are exactly twice. So it takes two hours to have one candle exactly twice as long as the other candle.
[[Link title]]

User:TanyaJacob

2010-09-22T03:44:25Z

TanyaJacob: Created page with 'Hi! This is Tanya. I'm originally from India but I have been living in Singapore for the past 7 years. I'm in first year Sauder :) '''The Pythagorean Theorem''' The Pythagore…'