## 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 area of the rectangle is a function of x, what value of x yields the largest area for the rectangle.

## Process

When given the problem we got a graphing paper and we had to try and plot the points of the points on the graph to get a vision of what the maximum rectangle we could draw inside of the parabola to be able to tell which points we can use that don't go outside of the parabola to find the maximum area. We then used the function we were given and created an x/y table to find out the points that we could plot on the graph.

## Solution

Solving for the maximum we had to have a function that we could use to find the maximum area. What we did was since A=LxW, and then since the length and width is X and Y, A=YxX. To find the maximum area of the rectangle the equation we used was A=-x^3+16x. With that equation we used a and X/Y table and with the equation we plugged in numbers to find the maximum area. We would haveto use decimals to get to the neareast area so we would have to go to the tenths and hundreds.

Solving for the permiter was a longer process than solving for the area depending on which you solved it. The function for permiter is P=2L+2WWith that function we used our original equation y=16-x^2 and plugged it in to the parenthesis. When plugging it in the equation would be P=2X+2(16-X). In order to solve for the the permiter we had to distribute across to get the final equation to get the answer. Once we distributed we got P=2X+32-2X^2. With the equation we had to create an X/Y table to find the permiter. We had to plug and chug a lot to try and get the exact number. Our final answer for the permiter was 32.5, and the X/Y axis points were (0.5,15.75).

Solving for the permiter was a longer process than solving for the area depending on which you solved it. The function for permiter is P=2L+2WWith that function we used our original equation y=16-x^2 and plugged it in to the parenthesis. When plugging it in the equation would be P=2X+2(16-X). In order to solve for the the permiter we had to distribute across to get the final equation to get the answer. Once we distributed we got P=2X+32-2X^2. With the equation we had to create an X/Y table to find the permiter. We had to plug and chug a lot to try and get the exact number. Our final answer for the permiter was 32.5, and the X/Y axis points were (0.5,15.75).

To solve for the maximum area we had to use the x/y table again to find the most precise plotting point. We went from using whole numbers to having decimals in or table to be able to find which one got be the closest to the maximum area. What we did to find the maximum area was using the original function and plugging in just numbers that were between two and three and with that we would just go more in depth and do 2.31 or 2.32 to get a more accurate answer.

## Group/ Individual Test

The day before the test Mr.Carter gave us the day to study for the test. So what he did was he gave another problem similar to the one we first got when we first got presented the problem. Our group we just followed the steps we used for the first problem on the the practice test problem. We made sure every step we did that each person in the group knew what we were doing and why so that when we did the test.

During the group test, I think our group did good. We all worked together to solve for the answers. What we did was just follow the same process of what we did with the other two problems we were presented. Along the way we were unsure of our answer because we felt it didn't make sense since the rectangle had to be in the first and second quadrant. But we were able to go back look over our steps and try to find out what we did.

As for the indivual test, I know I could've done better but I was rushing through to be done. I knew the steps and how to find the equations to solve for the area and perimeter. What I did was I knew how find the parabola to find where inside the parabola to know how big of the rectangle were working with. What I did when I was writing my answer was I wrote the wrong answer in and I got it wrong for not paying attention to the work I was doing.

Overall for the group quiz, our group did a good job working together solving for the area and permiter even though we weren’t sure of our answer we had a guess so we would leave our paper blank.

During the group test, I think our group did good. We all worked together to solve for the answers. What we did was just follow the same process of what we did with the other two problems we were presented. Along the way we were unsure of our answer because we felt it didn't make sense since the rectangle had to be in the first and second quadrant. But we were able to go back look over our steps and try to find out what we did.

As for the indivual test, I know I could've done better but I was rushing through to be done. I knew the steps and how to find the equations to solve for the area and perimeter. What I did was I knew how find the parabola to find where inside the parabola to know how big of the rectangle were working with. What I did when I was writing my answer was I wrote the wrong answer in and I got it wrong for not paying attention to the work I was doing.

Overall for the group quiz, our group did a good job working together solving for the area and permiter even though we weren’t sure of our answer we had a guess so we would leave our paper blank.

## Evaluation/Reflection

When we get problems like these introduced to us where we don’t know what step is going to come next after solving for one part really pushes your thinking. It makes you try and try to use previous methods to solve for the answer. If I had the opportunity to grade my self, I would give my self A because I really tried to solve for the answer with my group even though we were confused we didn’t give up, and whenever we needed help we would ask clarifying questions about our solution.