## Problem Statement

The

**problem read:***Maximum Rectangle**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

I came two days after the problem began, so I started it late. I began the problem in the middle of it when we were trying to find the perimeter of the rectangle. The equation we got for this problem was

**. We plugged in numbers from the equation that equation. This would give us points so we could plot them on a graph and create a parabola. We already had a given Y-Intercept of 16. Once we created the parabola, we had to find any rectangles inside of it and spot the largest one. The largest one had an area of 24 squares. We then used a guess and check method with 24 and the decimals in between. We created a equation to make it more simple. The equation went A = 16 - x^3 . The cube helped us to factor. We next needed to solve out for the perimeter. We made another equation. This time it was P = 2x^2 + 2x + 32 . We got that because you need to add all sides for the perimeter.***y = 16 - x^2*

## Solution

The Maximum Perimeter was 32.5 with side lengths of (0.5, 15.75). This was found by using our equation and turning it to vertex form. Once the vertex was found, we found that X is the width of it and Y is the perimeter.

The Maximum Area was between 21 and 24. We got 24.633609 with the equation we made. Our side lengths were 2.31 and 10.6639. We got them by rounding to the nearest hundredth.

The Maximum Area was between 21 and 24. We got 24.633609 with the equation we made. Our side lengths were 2.31 and 10.6639. We got them by rounding to the nearest hundredth.

## Group Test/Individual Test

For the group test, we got a practice one the day before. On this practice test, my group member Beto helped me to better understand the rectangle problem. We tried to finish as fast as we could while still getting the correct answers. We finished the practice test before class ended. The next day on the actual group test, we tried to repeat what we did the day before. The group test was a bit more difficult because it had two corners this time instead of one. The group test problem read:

*A rectangle has two corners on the graph of***y = 4 - x^2**, a third on the positive x-axis, and the fourth on the negative x-axis. The rectangle is symmetric about the y-axis and exists in the 1st and 2nd quadrants.**My group wasted no time and got started right away. It did not really feel like a test because my group was working the same as we always do with every problem we get in class. We all worked effectively with each other and each had something different to bring to the problem.

## Evalutation/Reflection

During this problem, I had to push myself a bit more to understand since I was missed out on the beginning of it. I did manage to catch up pretty fast though and understand what we had to being doing and how to be doing so. The thing that I was able to get most out of this problem was the time management that my group and I had during it. We worked very effectively and used our time wisely to just get the problem done with still getting a correct answer. If I were to give myself a grade on the overall problem, I would say a B+ because I did my part in my group and worked effectively but did not step up too much during it.