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?
Diagram (initial and final are same)
We needed to find the rectangle with the largest area that touched all of the mentioned corners. The diagram I have only includes the rectangles at whole numbers, but there are also an infinite amount of rectangles at decimal points.
Solution - Maximum Area
In order to solve for the maximum area, after we plotted the graph, we used our x&y to find the area of different rectangles. It was important to solve this problem efficiently, so we organized ourselves by first finding the area of wholexs, then moving on from the highest area, 2, to see if the vertex were somewhere between 1 and 2, or 2 and 3. We kept following this pattern until we got a more specific x value. The equation we used to find the area was a=x(16-x^2) which is just a variation of a=xy or a=lw. The answer was x=2.31
Solution - Maximum Perimeter
In order to solve for the maximum perimeter, after we plotted the graph, we used the equation p=2(x+(16-x^2)) or p=2x+2(16-x^2). This makes sense because perimeter for a rectangle is just p=2(l+w) or p=2(x+y), so we just elaborated on what y is. For this problem we followed the same process that we used to solve for area. We first found the perimeter of the whole xs, then got smaller between our highest number, 1, and the numbers next to it, to find a more specific decimal for our answer. Our solution was x=0.5
Group Test
To prepare for the group quiz, our table split up and worked individually on what we needed more help with. People in our group who were understanding the content better would help the people who were having trouble. These people would change depending on the topic, for example I needed some help with perimeter since I was absent, and other group members had some trouble organizing their tables so I could help with that.
I think that during the quiz, we did a good job of making sure the group was understanding the content while still be mindful of how much time we were taking. So for example when my group member did not understand the perimeter, we all sort of took turns trying to explain it quickly and answering questions so that we could move on with out leaving him behind. Personally I understood this problem very well. I got 30/30 on my individual test and the only problem our group got wrong on the group test was just a reading mistake that I quickly understood how to fix after seeing the highlighted portion of the question that we missed.
This group test was definitely one of the more challenging ones that we have done. The problem we did in class required less thinking and analyzing than the one we did for the quiz. I actually liked that it felt more like a puzzle and that we had to figure out when the time came. I think I definitely could have benefited by looking at more practice problems that were variations of the one we did in class.
Evaluation/Reflection
If I were to give myself a grade on this unit, I'd give myself an A or A+. I turned in all of my work and leave this problem feeling confident in the content we learned. This problem was mainly common sense. It reminded me a little bit of a logic riddle or something like that. I think I got the most out of learning how to organize my tables and work to be more efficient with my guessing and checking. To be honest, I wish we could have done a problem that would have allowed me to learn something deeper than this, but it is still an important skill to have.