Ferris Wheel Problem
Problem Statement
We started the ferris wheel problem off by trying to find the height of the diver when they were at different points on the ferris wheel. People solved this in many ways, but ultimately we all came up with the same formula to find the height of the diver at at any given point on the ferris wheel. This formula was 65+50sin9t, the 65 is the height from the center of the ferris wheel to the ground, 50 is the radius of the ferris wheel and you are trying to find the sin of degrees per second multiplied by the time because it's the vertical height. Next we went on to try to find the horizontal position if the diver on the wheel, this was easier for us since we had already found the height formula. The formula for this one was 50cos9t, in this case we were trying to find the horizontal position so that is why we use cos instead of sin, then we had the radius again and also the degrees per second. The next component we had to take into consideration for this problem was the tub that was moving at 15 ft/sec from a distance of -240 away from the ferris wheel. We had to match this up to the time the diver should be dropped from. We found this by seeing how fast the tub was moving compared to the distance that it was from the ferris wheel. The next thing and last to complete this problem was we had to do find the falling time of the diver and the formula that would help us easily find this was -240+15√ (W+ 57+50sin9W/16)=50cos9W. The -240 is the feet away from the ferris wheel that the tub starts from, and the 15 is the feet per second that the tub moves at. Inside the parentheses there's a W, which is the time it takes the wheel to move then there's the square root with a equation being divided by 16. The 57 in the equation is the height from the tub to the center of the ferris wheel then the 50 is the radius of the ferris wheel and we are finding the sin of 9W because are finding the height of the diver. This is all divided by 16 because the equation to find the time is originally h16. Then this all equals to 50cos9W because this is the equation to find the horizontal position of the diver.
Process
We took this problem step by step, every time adding a new element to try and figure out the whole entire problem. Some people around me tried right away to try to find the equations and others tried to just solve the question that was being given to us, but in the end we needed to find the equations whether that was finding them ourselves or having other people explain them to us. The last equation we had that combined all the previous ones was -240+15√ (W+ 57+50sin9W/16)=50cos9W. We started with just looking at the ferris wheel and the height of the diver and the speed of the ferris wheel, then we went to the horizontal position of the diver. After this we added the other element we had to take into consideration which was the tub, first we just found the position and speed of the tub, then in the end we connected this to the falling time of the diver to get our final answer.
Solution
We found that the object will fall 16t^2 in the first t seconds, then we add the height to this equation and we add this because we want to find out how far the diver is falling from their platform at any given time. After we get we add that we get f=√ h/16, we added this to our previous equations and that is how we got our solution. The solution to this problem is that the diver must be dropped at 12.2828 seconds in order for fall into the tub that is moving at 15 ft/sec from -240 feet away.
Assessment
I thought that this problem was very interesting and I did enjoy it because of the visual aspect. I am a person that learns through visuals, so the fact that we got to actually draw out the ferris wheel and picture the person falling into the tub helped me a lot. I believe that if we would have built the first ferris wheels we made to scale than this could have potentially helped many people, including myself, understand the problem even better.
Self Evaluation
I believe that I deserve an A, because I did do a lot of work for this problem, I think that I really pushed myself outside of my comfort zone by asking my peers questions when I didn't understand something. Also because we had to do a lot of analyzing and justifying in this problem, we had to back up our answers and we did this through the equations.
We started the ferris wheel problem off by trying to find the height of the diver when they were at different points on the ferris wheel. People solved this in many ways, but ultimately we all came up with the same formula to find the height of the diver at at any given point on the ferris wheel. This formula was 65+50sin9t, the 65 is the height from the center of the ferris wheel to the ground, 50 is the radius of the ferris wheel and you are trying to find the sin of degrees per second multiplied by the time because it's the vertical height. Next we went on to try to find the horizontal position if the diver on the wheel, this was easier for us since we had already found the height formula. The formula for this one was 50cos9t, in this case we were trying to find the horizontal position so that is why we use cos instead of sin, then we had the radius again and also the degrees per second. The next component we had to take into consideration for this problem was the tub that was moving at 15 ft/sec from a distance of -240 away from the ferris wheel. We had to match this up to the time the diver should be dropped from. We found this by seeing how fast the tub was moving compared to the distance that it was from the ferris wheel. The next thing and last to complete this problem was we had to do find the falling time of the diver and the formula that would help us easily find this was -240+15√ (W+ 57+50sin9W/16)=50cos9W. The -240 is the feet away from the ferris wheel that the tub starts from, and the 15 is the feet per second that the tub moves at. Inside the parentheses there's a W, which is the time it takes the wheel to move then there's the square root with a equation being divided by 16. The 57 in the equation is the height from the tub to the center of the ferris wheel then the 50 is the radius of the ferris wheel and we are finding the sin of 9W because are finding the height of the diver. This is all divided by 16 because the equation to find the time is originally h16. Then this all equals to 50cos9W because this is the equation to find the horizontal position of the diver.
Process
We took this problem step by step, every time adding a new element to try and figure out the whole entire problem. Some people around me tried right away to try to find the equations and others tried to just solve the question that was being given to us, but in the end we needed to find the equations whether that was finding them ourselves or having other people explain them to us. The last equation we had that combined all the previous ones was -240+15√ (W+ 57+50sin9W/16)=50cos9W. We started with just looking at the ferris wheel and the height of the diver and the speed of the ferris wheel, then we went to the horizontal position of the diver. After this we added the other element we had to take into consideration which was the tub, first we just found the position and speed of the tub, then in the end we connected this to the falling time of the diver to get our final answer.
Solution
We found that the object will fall 16t^2 in the first t seconds, then we add the height to this equation and we add this because we want to find out how far the diver is falling from their platform at any given time. After we get we add that we get f=√ h/16, we added this to our previous equations and that is how we got our solution. The solution to this problem is that the diver must be dropped at 12.2828 seconds in order for fall into the tub that is moving at 15 ft/sec from -240 feet away.
Assessment
I thought that this problem was very interesting and I did enjoy it because of the visual aspect. I am a person that learns through visuals, so the fact that we got to actually draw out the ferris wheel and picture the person falling into the tub helped me a lot. I believe that if we would have built the first ferris wheels we made to scale than this could have potentially helped many people, including myself, understand the problem even better.
Self Evaluation
I believe that I deserve an A, because I did do a lot of work for this problem, I think that I really pushed myself outside of my comfort zone by asking my peers questions when I didn't understand something. Also because we had to do a lot of analyzing and justifying in this problem, we had to back up our answers and we did this through the equations.
Simpson's Paradox
Problem Statement: The problem that we had to find our own solution to was Simpsons Paradox and answer the question “Is the application of the death sentence racially motivated?”. During this we evaluated three different graphs, one that had the total people who got the death sentence, one that had the total people who got the death sentence when the victim was white, and the third was the total people who were convicted when the victim was black. A lawyer from Florida was the one who collected all this data to try and prove his point that the death sentence application has been racially motivated. What the Simpsons Paradox is when a trend that appears in different groups of data but disappears or reverses when these two groups are combined. An example that I found of this is the US median wage decline, since 2000 the median wage has risen by 1% but over the same period within every educational subgroup the wage is lower now than in 2000. This is because there are now many more college graduates than there were in 2000, but wages for college graduates have collectively fallen and wages for those with even less education had fallen by even more. The growth in the portion of college graduates overloads the wage decline for specific groups.
Solving Process: To solve this problem I did look at the charts, but besides this I looked up the problem to get others evaluations and opinions on it. I saw that overall more white people who had been convicted got the death sentence and I think that this can be confusing to some people sometimes because of the claim being made. When you look farther into the graphs you see that 0 white people and only 3 black people who have been convicted have gotten the death penalty when the victim was black. This is very interesting because when the victim was white 39 white people and 32 black people got the death penalty. This meaning that the groups have averages that go in one direction while the overall average goes in the other. This is because the counts in subgroups differ between two populations.
My Solution: I belive that when people are convicted there is some racial motivation, and I think that it has both to do with the suspect at hand and also the victim. The data shows us that when theres a black victim there are 3 to none death penalties put into place, and it also shows us that when its a white victim people are convicted of the death penalty way more heavily. I think that the suspect racial application comes into play when you look at the 29 convicted white suspects and 0 were convicted of the death penalty against a black victim.
Other Students Process: The people that sat at my table during these discussions went through this same process, with analyzing the data and also sometimes using other sources and looking up the problem online. They agreed that both the suspects race and the victims race come into play when connoting a person of the death penalty. Explaining to me that you had to look at the graphs separately to really grasp the paradox concept, because if you looked at it as a whole you wouldn't think there was any racial biased. I think that a key part of me solving this problem was from my partners help because I wasn't at school the day that the assignment and problem was discussed.
Mr. Tejeras Solution: Our teacher Mr. Tejera told us that he also believed that the suspects and victims race come into play when a decision is being made. When you look at all the data it looks simple, that white people have gotten sentenced to the death penalty more, but in reality when you separate the data you see that a suspect is much more likely to get sentenced to the death penalty if they are black or if the victim is black instead of white.
Assessment of the Problem: I enjoyed this problem because of the information that we were reading about, the topic was very interesting to me besides having to solve it. I think that the question "Is the application of the death sentence racially motivated?" is what initially got me interested in the problem, from there I really did want to look at the data and make my own conclusions on what I thought. It was also very interesting to read about and hear other peoples points of view of the Simpsons Paradox.
Self Evaluation: I think that I did what I needed to for me to come to the conclusion I did, I researched when I needed to, and evaluated the data given to us. Aside from this I discussed the problem with my table mates to get their point of views. I think that I deserve and A on this assignment because I did try my best, and also used a lot of resources to solve the problem. So I collaborated to solve this problem and also looked at graphs/data to see patters in them and generalized this data. Next time I feel I will be even more ready to solve a problem like this because of all the information I have learned from this one and from out study of polls beforehand.
Journal Entry: I wrote a journal entry about my initial thoughts about Simpsons Paradox: I think that this is something that is very interesting and something that I definitely would like to go more in depth in. I will definitely look over the data and also try to decipher that more as well. I also want to ask more questions to my peers about this problem because I wasn't here the first day that the problem was given.
Solving Process: To solve this problem I did look at the charts, but besides this I looked up the problem to get others evaluations and opinions on it. I saw that overall more white people who had been convicted got the death sentence and I think that this can be confusing to some people sometimes because of the claim being made. When you look farther into the graphs you see that 0 white people and only 3 black people who have been convicted have gotten the death penalty when the victim was black. This is very interesting because when the victim was white 39 white people and 32 black people got the death penalty. This meaning that the groups have averages that go in one direction while the overall average goes in the other. This is because the counts in subgroups differ between two populations.
My Solution: I belive that when people are convicted there is some racial motivation, and I think that it has both to do with the suspect at hand and also the victim. The data shows us that when theres a black victim there are 3 to none death penalties put into place, and it also shows us that when its a white victim people are convicted of the death penalty way more heavily. I think that the suspect racial application comes into play when you look at the 29 convicted white suspects and 0 were convicted of the death penalty against a black victim.
Other Students Process: The people that sat at my table during these discussions went through this same process, with analyzing the data and also sometimes using other sources and looking up the problem online. They agreed that both the suspects race and the victims race come into play when connoting a person of the death penalty. Explaining to me that you had to look at the graphs separately to really grasp the paradox concept, because if you looked at it as a whole you wouldn't think there was any racial biased. I think that a key part of me solving this problem was from my partners help because I wasn't at school the day that the assignment and problem was discussed.
Mr. Tejeras Solution: Our teacher Mr. Tejera told us that he also believed that the suspects and victims race come into play when a decision is being made. When you look at all the data it looks simple, that white people have gotten sentenced to the death penalty more, but in reality when you separate the data you see that a suspect is much more likely to get sentenced to the death penalty if they are black or if the victim is black instead of white.
Assessment of the Problem: I enjoyed this problem because of the information that we were reading about, the topic was very interesting to me besides having to solve it. I think that the question "Is the application of the death sentence racially motivated?" is what initially got me interested in the problem, from there I really did want to look at the data and make my own conclusions on what I thought. It was also very interesting to read about and hear other peoples points of view of the Simpsons Paradox.
Self Evaluation: I think that I did what I needed to for me to come to the conclusion I did, I researched when I needed to, and evaluated the data given to us. Aside from this I discussed the problem with my table mates to get their point of views. I think that I deserve and A on this assignment because I did try my best, and also used a lot of resources to solve the problem. So I collaborated to solve this problem and also looked at graphs/data to see patters in them and generalized this data. Next time I feel I will be even more ready to solve a problem like this because of all the information I have learned from this one and from out study of polls beforehand.
Journal Entry: I wrote a journal entry about my initial thoughts about Simpsons Paradox: I think that this is something that is very interesting and something that I definitely would like to go more in depth in. I will definitely look over the data and also try to decipher that more as well. I also want to ask more questions to my peers about this problem because I wasn't here the first day that the problem was given.
The Birthday Paradox
11/1/15
Problem Statement:
The question that we had to solve for was: How many people do you need in a group for the probability that two people will have the same birthday to become 50%?. We had to look at this problem in a critical way, because we wouldn't just have to find a the answer to the problem, but also a formula so that if we wanted we could find other probabilities and percentages.
The Process:
When given this problem the first thing I did was try to find the solution, even if that meant doing it a longer way, instead of maybe finding a formula that could lead me to solving the problem. So I just started finding the different percentages for the 365 days of the year by multiplying 364/365*363/365*362/365 and so on until getting to 50%. That is what lead me to the answer which is 23 people, it takes 23 people for there to be a 50% chance that two people will have the same birthday in that same group. Knowing that there was a much easier way to solve this problem I started trying to figure out a formula.
Other's Process:
My table mates also took this same approach to trying to solve the problem, and they also got the same anwser. After this is when we started to try to figure out a formula. I struggled with trying to find a formula that would work and turned to my table mates for some help. When they were struggling as well we turned to Mr. Tejera who gave us some advice, he suggested that we use a factorial in our formula. A factorial is the product of the integer and all the integers before it, this would save us from having to individually multiply all these numbers like we had done before.
The Solution:
I ended up finding the answer to the problem which was you need 23 people in a group for the probability that two will have the same birthday to become 50%. I still couldn't figure out a formula that we could have used instead.
Mr. T Solution:
In the end Mr. Tejera gave us his solution to the problem which was 365!/365^n(365-n)!, "n" represents the number of people, so if you were to plug in 23 then you would get 50% in return.
Assessment of Problem:
I enjoyed the concept of this problem I had never really thought of that it is really likely that someone may have the same birthday as you, that might be because I am a twin so that has happened naturally to me, but the probabilities are high that two random people might have the same birthday in a small group. I think that once we got into trying to find the formula this is where I had to push my self a little more, because it wasn't easy to find. Talking with my table mates and getting their support definitely helped me.
Self Evaluation:
I think that I deserve a A because of my effort throughout the problem, and I did find the anwser just not the complete solution. Also when I needed help or didn't understand something I asked my class mates for help, instead of just giving up. This is an area that I have grown in math, because I usually don't like asking for help in math, but now I have realized that there is nothing wrong with asking questions.
The question that we had to solve for was: How many people do you need in a group for the probability that two people will have the same birthday to become 50%?. We had to look at this problem in a critical way, because we wouldn't just have to find a the answer to the problem, but also a formula so that if we wanted we could find other probabilities and percentages.
The Process:
When given this problem the first thing I did was try to find the solution, even if that meant doing it a longer way, instead of maybe finding a formula that could lead me to solving the problem. So I just started finding the different percentages for the 365 days of the year by multiplying 364/365*363/365*362/365 and so on until getting to 50%. That is what lead me to the answer which is 23 people, it takes 23 people for there to be a 50% chance that two people will have the same birthday in that same group. Knowing that there was a much easier way to solve this problem I started trying to figure out a formula.
Other's Process:
My table mates also took this same approach to trying to solve the problem, and they also got the same anwser. After this is when we started to try to figure out a formula. I struggled with trying to find a formula that would work and turned to my table mates for some help. When they were struggling as well we turned to Mr. Tejera who gave us some advice, he suggested that we use a factorial in our formula. A factorial is the product of the integer and all the integers before it, this would save us from having to individually multiply all these numbers like we had done before.
The Solution:
I ended up finding the answer to the problem which was you need 23 people in a group for the probability that two will have the same birthday to become 50%. I still couldn't figure out a formula that we could have used instead.
Mr. T Solution:
In the end Mr. Tejera gave us his solution to the problem which was 365!/365^n(365-n)!, "n" represents the number of people, so if you were to plug in 23 then you would get 50% in return.
Assessment of Problem:
I enjoyed the concept of this problem I had never really thought of that it is really likely that someone may have the same birthday as you, that might be because I am a twin so that has happened naturally to me, but the probabilities are high that two random people might have the same birthday in a small group. I think that once we got into trying to find the formula this is where I had to push my self a little more, because it wasn't easy to find. Talking with my table mates and getting their support definitely helped me.
Self Evaluation:
I think that I deserve a A because of my effort throughout the problem, and I did find the anwser just not the complete solution. Also when I needed help or didn't understand something I asked my class mates for help, instead of just giving up. This is an area that I have grown in math, because I usually don't like asking for help in math, but now I have realized that there is nothing wrong with asking questions.