Problem Statement- A rectangle has one corner on the graph of y= 16 - x², another a the orgin, a third on the positive y-axis and the forth on the positive x-axis. If the ara of the rectangle is a function of x, what value of x yeilds the largest area for the rectangle?
Process initial attempt- The first thing that my table group decided to do was make a x,y table and try to plot points into the graph that was given to us.to find the points on the graph we used the equation y= 16 - x² and plugged in the numbers on the x axis. (Diagram 1). we then ended up with a parabola (Diagram 2)
we then made a generalized function that will later on help us find both maximum area and maximum perimeter (diagram 7)
We drew in rectangles to find the maximum area (Diagram 3)so in order to find the maximum area we used the equation A= (x × y) and found that points (2, 12) had the highest area of 24 (diagram 4).
In order to find the maximum area we had to look in-between points 2 through 3. stating with 2.1 (diagram 5) we found that 2.3 was the highest (because every other point between 2 and 3 would decrease)., giving us our max area of 24.63
we then had to find out the max X value which will later on help us find out the max perimeter. So from 2.3 we went deeper and found out the area of the point 2.31. making our max X value equal 24.63193
using the generalized function that we made in the beginning for the perimeter. we will plug in y= 16 - x² into 2(x + y) to find out our maximum perimeter. making our final equation 2 ( x + 16 - x²). plugging in 0.5 into the x value we solved what was in the parenthesis, getting 16.25, then multiplying that number by 2. resulting in our maximum perimeter of 32.5
Solution maximum perimeter after finding the maximum area we then moved into finding the maximum perimeter. but first we needed to find the maximum x value so we made another table that was in between 2.3. We were able to find out that 2.31 is going to be the highest x value, equaling 10.633 and by using the (x ×y) function we were able to find the max x value of 24.63193.
so then we used the new function 2 ( x + 16 - x²) to find the max perimeter. We plugged in 0.5 into the x value and solved what was in the parenthesis getting our maximum peremeter as 32.5 maximum area my groups final solution was 24.63. we were able to get this final answer by finding the plot point with the highest area (2, 12) with an area of 24. and we looked deeper into the number 2. using the equation 16 - x² we plugged in numbers going up by 2.1, 2.2 , (etc.) We found that numbers after 2.3 were decreasing. which meant that 2.3 was the highest. So taking that y point 10.71 we used the function (x ×y) we then got our maximum area of 24.63
group test/ individual test description of how my group prepared for the group quiz for the practice group test my group didn't know how to prepare at first so we went back to the original problem to see what we did overall. So with the new problem presented to us we were able to organize what we needed to do step by step by finding the points to finding the max area so we can move onto the max perimeter.
how I felt my group worked to understand the problem during the group quiz I think that my group did a good job on getting the work done but I think that there was a lack of communication between some people which resulted in other group members not being caught up or being too far ahead of others. But after we realized that we weren't working together very well, we kept telling each other "this is a group test" when someone would go off on their own so we can focus as a group.
description of my ability to understand the individual test During the individual quiz knew what steps to do but I didn't read the question carefully and ended up doing something completely different from the question. But I do think that if my group was able to have better communication I would have been able to find the perimeter when x= 1.1. because I understand what to do until it gets to finding out what the maximum perimeter is.
overall experience of the group quiz I think that the overall experience of the group quiz was pretty frustrating because we got off track way too often from trying to work tougher because others would be lost or would be way too ahead of where everyone else was and wouldn't know what part we were working on which resulted in small arguments over the different answers we've gotten or not being Able to explain why one way was right or not.
evaluation/ Reflection Something that pushed my thinking during this problem was trying to find ways to be helpful to my group since we were all confused as what we should do to start. So I asked them questions that would lead to our group collaborating to go further into the problem. This unit I was able to see that one other group member wasn't contributing as much so I thought that they might not understand what the rest of the group was dong, which lead me to asking them what they were confused about and to explain the problem back to me so they were able to understand the problem instead of copying down what we were writing.. If I were to grade myself on this unit I think I would give myself an A. comparing it to the cow problem I think I said that I was trying to improve on posing my own ideas to the group and I think I did that a lot more this time and by working with another person to (somewhat) lead our group to stay on the right track helped out a lot with my own understanding of what we needed to do to finish the problem.