Problem statement
2
A rectangle has one corner on the graph of y=16-x , 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?
A rectangle has one corner on the graph of y=16-x , 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?
When we were first presented to this problem I tried to draw what I thought it would look like, then my group decided to do a X and Y table. We plugged in numbers 1-4 to y and then from there we realized that our largest number for x will be 4. We graphed it and we got a bunch of rectangles which determined that those points where going to determine the maximum area and perimeter.
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Coming into this problem I was confused on what we had to do. I was very confused and I would just plug in the numbers that my group mates would tell me to plug in for the functions. Eventually I started asking questions and started understanding the problem more and started being able to find the maximum area and perimeter on my own.
Maximum Area
Next what we had to do was find the maximum area. The function for this problem would be A=Length x Width Length being X and width being Y which becomes. 2 A= x(16-x ) After you distribute the x it becomes 3 A=16x-x We knew that our maximum area would be in between 2 and 3 because we plugged in numbers through 0-4 and 2 and 3 were the highest. My group and I started plugging in numbers with decimals like 2.1, 2.2, 2.213 , etc. Eventually after what we call "plug and chug" we came to a conclusion that 2.3 was the highest maximum area. |
Maximum Perimeter
Next what we wanted to find was the maximum perimeter. The function for the maximum perimeter is 2 2(x+y ) Which would turn out to be 2 2(x+16-x ) In order to find the maximum we tested out numbers 0-4 and got the 2 highest numbers and then started plugging in decimals eventually we found out that 0.5 was the highest maximum area. |
After being able to solve for the area and the perimeter each table had to write their final answers for the problem. Once we all finished most of the groups went on and presented their own. Everyone had the same answer for the perimeter and area which let us know that we were correct. Mr.Carter went over it as well, which helped better our understanding of the problem.
Final answer Our final answer for our maximum area is 24.63 with the side lengths of (2.3,10.66). Our final for the maximum perimeter is 32.75 with the side lengths of (0.5,15.75). |
Group/Individual Test
My group and I prepared for the group test by practicing on the same problem Mr.Carter gave us just with a different number. My group and I were able to solve it quickly and then Mr.Carter gave us a challenge which was, what would happen if it was on both sides of the graph? My group and I discussed it and wondered if it was going to be the same numbers just on the negative axis . I believe that my group and I did really good on the test. We all knew what we were doing,. I was aware of the steps were in order to find the maximum numbers. My group and I collaborated really well together. For the individual test, I got stuck at first, but then I realized how easy it was. The value of X was given to us, all we had to do was just plug it in and get the answer. I plugged in x for the area, but I was afraid that it was going to be a trick question so I plugged in 2.2,2.3,2.4 , etc. For the perimeter I didn't have much time to finish it, but I plugged in X, but made a simple mistake by not multiplying it by 2. After discussing it with my peers , I realized how easy it was. Overall I am very proud of my performance in the group/individual test, because I got a good grade on it. It was a good math experience, even though at first I didn't get it.
Reflection
This problem really pushed my thinking. I wish I had participated more in the beginning .When we first got introduced to this problem I did not understand it and I didn't ask my group questions because I was afraid . Eventually I asked questions and started understanding what was being asked in the problem. I started participating more and was very interested in being able to understand it. Something that I got out of this was that I could understand math if I try really hard and study hard. Overall if I were to grade myself on this unit I would give myself a B+ because I know I could've pushed myself a lot more in the beginning. Towards the end is when I started pushing myself more on understanding this unit.
My group and I prepared for the group test by practicing on the same problem Mr.Carter gave us just with a different number. My group and I were able to solve it quickly and then Mr.Carter gave us a challenge which was, what would happen if it was on both sides of the graph? My group and I discussed it and wondered if it was going to be the same numbers just on the negative axis . I believe that my group and I did really good on the test. We all knew what we were doing,. I was aware of the steps were in order to find the maximum numbers. My group and I collaborated really well together. For the individual test, I got stuck at first, but then I realized how easy it was. The value of X was given to us, all we had to do was just plug it in and get the answer. I plugged in x for the area, but I was afraid that it was going to be a trick question so I plugged in 2.2,2.3,2.4 , etc. For the perimeter I didn't have much time to finish it, but I plugged in X, but made a simple mistake by not multiplying it by 2. After discussing it with my peers , I realized how easy it was. Overall I am very proud of my performance in the group/individual test, because I got a good grade on it. It was a good math experience, even though at first I didn't get it.
Reflection
This problem really pushed my thinking. I wish I had participated more in the beginning .When we first got introduced to this problem I did not understand it and I didn't ask my group questions because I was afraid . Eventually I asked questions and started understanding what was being asked in the problem. I started participating more and was very interested in being able to understand it. Something that I got out of this was that I could understand math if I try really hard and study hard. Overall if I were to grade myself on this unit I would give myself a B+ because I know I could've pushed myself a lot more in the beginning. Towards the end is when I started pushing myself more on understanding this unit.