Monte Hall Gameshow Blog: Response

The Birthday Problem:
To solve the birthday problem, you need to find the probability that two people don't have the same birthday. Then you use the compliment (1-P(two people don't have the same birthday)). Then because of exponential growth, the chances of getting a match increase. Exponential growth rapidly decreases the chance of two people not having the same birthday, and therefore the chances of having the same birthday increases. The reason that when you have 23 people, the chances of having the same birthday are about 50% because the square root of 365 is about 20, meaning that about 50% of the time when there are about 20 people, two people will have the same birthday.

Using Semi-Log Graphs

Graphs on the semi-log paper are more linear, than those on regular graph paper. They also are to a much different scale than graphs on rectangular graph paper because the vertical axis increases by exponential powers. 

The main reason to graph data on semi-log graph paper is that when dealing with logs and exponential functions, the number increase rapidly, and semi-log paper allows you to plot all of your points, instead of only being able to plot 2 or 3 like on rectangular graph paper. Basically, if the graph is increasing exponentially, then using semi-log graph paper is easier. 

One advantage to using semi-log graph paper is that you are able to graph exponentially increasing functions that have large numbers. One disadvantage is that you can't really see the shape of the graph because semi-log graph paper produces a more linear looking line. 

A graph that has a logarithmic scale, and that has been graphed on semi-log graph paper could be misleading because the scale on semi-log graph paper is much different. It condenses the plot so that you can fit all of your points on the paper. 

The Significance of Exponential Functions (Monica Blog)

The main thing I learned from Monica's talk was that information spreads quicker than any of us can imagine. As the negative information spreads, and more and more people see it, life becomes less worth living for the victim. When she was talking about how many people who are victims of cyberbullying commit suicide, that's when her message really hit me, and made me realize how serious this problem is. 

If every hour, 3 new people get the information, then the chart looks like this:
x     y
0     1
1     3
2     9
3     27
4     81
5     243
6     729
7     2187
8     6561
9     19683
10   59049
11   177147 
12    531441
13    1594323
14    4782969
15    14348907

The equation of these data points is y=3^x
When you try to graph these points on the graph paper, you can only do the first three because after that, the number increase my too much and are therefore too large to graph on the paper.

If every hour 10 new people get the information, then the chart looks like this:
x     y
0     1
1     10
2     100
3     1000
4     10000
5     100000
6     1000000
7     10000000
8     100000000
9     1000000000
10   10000000000
11   100000000000
12   1000000000000 
13   10000000000000
14   100000000000000
15   1000000000000000

The equation of these data points is y=10^x
With these points, the numbers increase so rapidly that on the graph, you can only plot the first two points. 

If the information was passed on to 10 people every 15 seconds then after only one minute, 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 would have the information. This number is so large that it is hard for us to grasp that concept of how many people are receiving information per minute if only 10 people receive it every 15 seconds. (y=10^60)

If the information was passed on to 100 people every time someone "clicks", the number after only one minute would be way too large for us to imagine.

Every time you increase the number of people receiving the information, the graph gets steeper. For example the graph of y=10^x is much steeper than the graph of y=3^x. This is because as you increase the base, the numbers you get when you plug in a number for x get drastically bigger. 

In conclusion, it is impossible for us to grasp just how many people are seeing information on the internet per minute, or even second.

Math Humor

I got all of these pictures off of pintrest, but they are also cited individually from the actual website that they came from.

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Proving Trig Identities: Response

a) The equation tan(x)^2+cos(x)^2=sin(x)^2 is not a trig identity. We can verify that by putting it into the calculator, and looking at both how the graph of the rhs and lhs do not match up, and by looking at the table to see that the y1 and y2 columns don't match up.




















b) The equation 1-sec(x)^2=-tan(x)^2 is a trig identity. We can verify, but not prove this using the graph on our calculator.





















c) The reason that you can only use a calculator to verify a trig identity, and not prove it is because your calculator cannot do the algebra. While you can look at the graph to verify whether you have solved the identity, your calculator cannot show you the algebra in getting to the answer.

Without Trig, Your House Would Be Lopsided: Response

Finding the angle of elevation:
tanx=11/18

tan^-1(11/18)=x

x=31.430 degrees

So, the angle of elevation is 31.430 degrees.

Pitch, grade, and slope are all the same values on the triangle. This is also the hypotenuse value. All of these terms are showing the steepness or incline of the hypotenuse of the given triangle.

Tangent is also related to pitch, slope, and grade because you use tangent to find the angle of elevation, which is used to find the pitch. Also, when you use tangent, you are taking the tangent of the slope.

The total rise is 22 inches because, for every 18 feet horizontally, there is 11 feet vertically. Looking at the image, there are 2, 18 ft. horizontal runs, so that means that you multiply 11 x 2, to get 22 in.

The full length of the rafter line is found using the equation L=sqaure root of (1.8) squared + (22) squared. This gives us L=22.074 ft. which is the full length of the rafter line.

Err In the Direction of Kindness Response

Hi Ms. Mariner,
Thank you for sharing this blog with us! I think it is a good reminder that everybody needs to step back sometimes, especially during a time of high emotion (good or bad), and reflect on how they can be more accepting of society, and the people around them. Sometimes, we are so critical of ourselves, and try to change who we are, so that we will fit in with society. Saunders provided us with a powerful speech that invokes hope in an otherwise somewhat hopeless world. It just goes to show that you will regret many things in your lifetime, but the one thing many people regret the most is not being kind enough to others. This article is a good reminder that we all get caught up thinking about ourselves, but in the end, it is the world as a whole that matters, and not just yourself.