Problem Statement
A rectangle has one corner on the graph of y = 16 - x^2, another at the origin, a third on the positive y axis, and the fourth on the positive x axis. If the are of the rectangle is a function of x, what value of x yields the largest area for the rectangle?
Process
First attempt:
When we were first assigned this problem we were asked to try things out in attempt to figure out the answer. We were told that if we didn't know how to approach it, we should write down question, which is what I did. I wanted to know what exactly I was supposed to be looking for and what skills I already had should I use to find the answer. After a class discussion we found out that the equation would end up graphing a parabola. I drew a parabola on a graph (without any points) and drew a rectangle within it according to the description in the problem just to understand the general shape of the rectangle we were finding the area of. |
Final diagram:
For our final diagram we plugged in the numbers 0-4 as the x in order to find the y so that we could plot the points on the graph. When we plotted these points it gave us an accurate shape for our parabola, where we then drew in the rectangle following directions in the problem. This gave us the correct diagram so we could start looking for the area.
For our final diagram we plugged in the numbers 0-4 as the x in order to find the y so that we could plot the points on the graph. When we plotted these points it gave us an accurate shape for our parabola, where we then drew in the rectangle following directions in the problem. This gave us the correct diagram so we could start looking for the area.
How we found Area:
What we first did to find the area was multiply the x and the y. We did this with numbers 0-4 as well. The number with the highest area was 2, which had an area of 24. once we found which number had the highest area, we looked for the tenths and the hundreds with the highest area. We knew that this number had to be in between 1 and 3. In order to do this we used the formula A=x(16-x^2) and doing what was in the parenthesis first. We tried 1.9 first which was less than 24 so that means we didn't have to do anything that comes before that because it would be to small. The value of x in the tenths that has the highest area is 2.3 which has an area of 24.633. The value of x in the hundreds that has the highest area is 2.31 with an area of 24.6336. |
Perimeter:
To find the perimeter I just did Length+length+width+width. To find the width you subtract the x from 16. The highest perimeter we got was 32. We then moved on to the tenths and did the same thing for this I got 31.78 with the side lengths of 1.1 and 14.79. When I did the hundreds, no matter which side lengths I tried they all gave me the perimeter of 32. I don't know if I did anything wrong but it was odd. |
Solution
The value of x that yields the highest area for the rectangle in the tenths place was 2.3, the area was 24.633. The Value of x that yields the highest area for the rectangle in the hundreds place was 2.31, the area was 24.6336.
The highest perimeter was 31.78 and the side lengths where 1.1 and 14.79.
The highest perimeter was 31.78 and the side lengths where 1.1 and 14.79.
Group/individual quiz
Before the group quiz:
My group prepared for the quiz by working on this problem to try to understand it more deeply as well as trying it out with different numbers instead of 16 (the original problem) to be prepared for the fact that on the test it was going to be a different number.
During the group quiz:
I'm not sure how my group mates felt during the quiz, but I felt like we didn't work together to understand the problem as well as we could have. I personally felt ignored when I shared my opinions and thoughts because other group mates were convinced their ideas where the only ones that could be right. I think we could have worked better together if we could have all listened to each other and asked questions as to why we believed we should solve it in a certain way.
During the individual quiz:
I'm not going to lie I don't think I performed as well as I wish I had on the individual quiz. I thought I knew what I was doing but I guess sometime along the way I must have forgotten how to do a piece of the work. While we were working on this in class I felt like I understood it and didn't feel like I had a problem. As soon as I saw the questions on the test it was as if all of the things I thought I knew about this topic flew away from my memory. Honestly, I do feel like I put in as much effort as I could to understand the work but I don't know what happened. I want to continue to work on this content even though the problem is over because I want to be able to know what I did wrong on the text and to understand why it was wrong and how to fix it.
Overall group quiz experience:
Even though my group work experience wasn't the best in my opinion, I do appreciate it. It genuinely was a learning experience for me. I know that you're not always going to get to work with people you know and like. On occasion you will get put in a group that you don't necessarily mesh well with and you might have to learn how to deal with it. Even though it's hard, if I ever feel this way in a group I will speak up and not let myself get ignored because my ideas matter as well.
My group prepared for the quiz by working on this problem to try to understand it more deeply as well as trying it out with different numbers instead of 16 (the original problem) to be prepared for the fact that on the test it was going to be a different number.
During the group quiz:
I'm not sure how my group mates felt during the quiz, but I felt like we didn't work together to understand the problem as well as we could have. I personally felt ignored when I shared my opinions and thoughts because other group mates were convinced their ideas where the only ones that could be right. I think we could have worked better together if we could have all listened to each other and asked questions as to why we believed we should solve it in a certain way.
During the individual quiz:
I'm not going to lie I don't think I performed as well as I wish I had on the individual quiz. I thought I knew what I was doing but I guess sometime along the way I must have forgotten how to do a piece of the work. While we were working on this in class I felt like I understood it and didn't feel like I had a problem. As soon as I saw the questions on the test it was as if all of the things I thought I knew about this topic flew away from my memory. Honestly, I do feel like I put in as much effort as I could to understand the work but I don't know what happened. I want to continue to work on this content even though the problem is over because I want to be able to know what I did wrong on the text and to understand why it was wrong and how to fix it.
Overall group quiz experience:
Even though my group work experience wasn't the best in my opinion, I do appreciate it. It genuinely was a learning experience for me. I know that you're not always going to get to work with people you know and like. On occasion you will get put in a group that you don't necessarily mesh well with and you might have to learn how to deal with it. Even though it's hard, if I ever feel this way in a group I will speak up and not let myself get ignored because my ideas matter as well.
Evaluation
I really did try my hardest to understand this problem. I personally feel like even though I had a hard time on the quiz, I did learn a lot. While looking over my notes to be able to write this write up, I felt a lot more comfortable with this topic and honestly feel like I could do much better if I were to re-do this quiz. For effort, I would honestly give myself an A. During class I never gave up and asked my group mates questions if I needed help. After the quiz I went back and looked at all my work to figure out where I went wrong on the quiz and used what I found out to write my write up. Even though I didn't do my best on the quiz, I am very proud that I learned from my silly math mistakes and kept on trying the problem until I gained a deeper understanding of it.