Problem Statement
A rectangle has one corner on the graph of (y=16−x²), another at the origin, a third on the positive yaxis, and the fourth on the positive xaxis. If the area of the rectangle is a function of x, what value of x yields the largest area for the rectangle.
Process
In our initial attempts we had to find perimeter and area for this rectangle using what we have which is y=16x². We had to find formulas for area and perimeter to find the maximum of each. We knew that perimeter=2l+2w, and that area=x * y. We were given "y", so we plug the formula of "y", into these perimeter and area equations. We plugged it in and simplified it to its simplest form. After we need to find "x", by plugging in different numbers starting at 0. We plugged in "x", and we were looking for what gave us largest number because that we equal the maximum area, the equation for this was x³+16. Then started plugging into the perimeter and the equation looked like this, 2x^2+2x+32. We plugged in numbers into these equations starting from zero and when we started noticing that when we got to certain number and it started decreasing, we knew that the maximum point was near the decreasing number.
Solution
Maximum Area:
To find the maximum area, you first have to know what the equation for finding area is. The equation for finding the area of a rectangle is (Area= length times width). We don´t have length and width with the rectangle that we´re given because we´re graphing it on the positive x and y axis. In that case, we´re going to need to switch up the formula to look like this, (Area= x times y). Now that we have our adjusted Area function, we can plug in the Y that was given to us in the problem statement, (Y=16x²) so our equation should look something like this, A= x(16x²). We can distribute the x to the 16 and the x² so our simplified equation would look like this,(A=16xx³).
Maximum Perimeter:
To find the maximum perimeter, you first have to know the equation to find Perimeter. The equation for Perimeter is P=2l+2w. Now, in our case we had to plug in x and y so we just replaced l and w with x and y (P=2x+2y). Now that we have our adjusted Perimeter function, we can plug in our y (Y=16x²), which would look something like this, P=2x+2(16x²). Now that we have our y plugged in, we can distribute the 2 to the 16 and x² which would look like this, P=2x+322x². Now if you have Desmos you can plug in and get your maximum perimeter but in our class that was not our case. We had to plug and chug a lot to find the maximum perimeter using a t chart to find the maximum perimeter. Now if you did all that and found the x and y axis points (0.5, 15.75) that give you the maximum perimeter (32.5) you have successfully solved for perimeter.
To find the maximum area, you first have to know what the equation for finding area is. The equation for finding the area of a rectangle is (Area= length times width). We don´t have length and width with the rectangle that we´re given because we´re graphing it on the positive x and y axis. In that case, we´re going to need to switch up the formula to look like this, (Area= x times y). Now that we have our adjusted Area function, we can plug in the Y that was given to us in the problem statement, (Y=16x²) so our equation should look something like this, A= x(16x²). We can distribute the x to the 16 and the x² so our simplified equation would look like this,(A=16xx³).
Maximum Perimeter:
To find the maximum perimeter, you first have to know the equation to find Perimeter. The equation for Perimeter is P=2l+2w. Now, in our case we had to plug in x and y so we just replaced l and w with x and y (P=2x+2y). Now that we have our adjusted Perimeter function, we can plug in our y (Y=16x²), which would look something like this, P=2x+2(16x²). Now that we have our y plugged in, we can distribute the 2 to the 16 and x² which would look like this, P=2x+322x². Now if you have Desmos you can plug in and get your maximum perimeter but in our class that was not our case. We had to plug and chug a lot to find the maximum perimeter using a t chart to find the maximum perimeter. Now if you did all that and found the x and y axis points (0.5, 15.75) that give you the maximum perimeter (32.5) you have successfully solved for perimeter.
Group Test
Preparing for the group quiz was a lot of reviewing. Obviously to be very prepared for a test you study and review what you studied over and over again. That’s exactly what we did as a group. We took each other’s work and we all looked over it together to see if as a group we were on the same page in progress, as well as the same understanding of each concept.
During the group quiz we all felt good heading into it, I was personally nervous because I still get uncertain with graph work even though I have studied. We really communicated throughout the whole quiz. There wasn’t a time where we left someone behind, there was times where myself and peers got stuck, but we explained and guided one another. Very positive work with each other. 
Individual Test
During the Individual test is where me as a student and my learning would be showcased throughout this two week span of learning this problem. Coming off of the group quiz, I went into the individual quiz feeling really good, it was all thanks to my peers. My understanding of the individual quiz was better than expected.

Experience Throughout Problem
The whole experience was really good for me because this math concept was one I really needed to work on and get better at. I did just that throughout this problem which to me is a huge success!
Evaluation/Reflection
Honestly the work with a graph is what really pushed my thinking throughout this math problem. I am not very good at working on graphs, and that shows on some previous math tests i've took. The graph work in this math problem was like nothing I have encountered before. Working on graphs is what I really took the most out of while working on this. It was difficult for me to understand at first, but when I got a little boost from my peers, I understood everything after. I contributed to the thinking, by confirming and suggesting different numbers in formulas, it felt good understanding the graph work. I feel like I deserve an A on this project because of how I got through the personal struggle of graph work. I hope to develop myself with working with graphs a lot more.