## The Problem

In this problem we are asked to find a rectangle that produced the largest area and a rectangle that produced the largest perimeter, but that fit these givens.

The rectangle must have:

The rectangle must have:

- One corner at the origin, point (0,0)
- One corner on the positive Y Axis
- One corner on the positive X Axis
- One corner on the on the graph of y= 16 - x^2

## The Process & Solution

The first thing my group did was try to draw out the problem. We graphed the function we were given by creating a table and then plotting the points. This made visualizing the rectangle much easier. We now knew that one corner of the rectangle would fall somewhere on the pink line and the other two on the green lines. This meant the side on the y-axis could between 1-15 and the x-axis, between 1-4.

Next we started looking for the rectangle that would give us the largest area. We started by using the basic area formula that we knew Area = Base X Height. We then manipulated the formula to fit our problem. The base of our rectangle was represented by x and and the height by y. We were again able to substitute y because we were given y = 16 - x^2. Our final formula was Area = x (16 - x^2).

Next we created a new table and began plugging in x values into our formula. When we finally found the rectangle with the side lengths of 2 and 12 gave us the largest area, we decided to narrow our search and try decimals between 2 and 3 to see if any were bigger. NEED IMAGE OF X/Y/A TABLE

Again we used our formula to create a table.

## Group & Individual Test

I prepared for the group quiz by working with my team to solve a similar problem as the one we studied. In the problem we again had to find the largest area and perimeter but for a rectangle with a corner on the graph of y = 25 - x^2. We just practiced and explained the steps we took to ensure our understanding. I didn't have a very positive experience taking the group test because I felt that the test was quite different than the problem we studied. Despite being a little caught off guard, our group worked very well together. The individual test went much better for me. I was easily able to answer the questions using the formulas we had discovered. I would say my overall experience with the group quiz was okay. Immediately leaving the test I was a little worried how severely the group quiz would effect my grade. I think the hardest part of this test was not really having anything to study. I know I perform a lot better and learn more when we have lecture based units.

## Evaluation & Reflection

I really enjoy math, so getting to work on any problem is fun for me. This time, I do think I had a hard time transitioning from the previous unit which was lecture and exam based to this more collaborative learning. I think working with my group pushed me in a different way than usual. I am usually able to challenge myself by attempting more advanced problems and working at a much faster pace, but this unit I had to slow down to help out my team mates. Both are important skills that I have improved this semester. If I were to grade myself I would give myself an A- because I was an attentive and collaborative group member but I didn't really take initiative in challenging myself academically.