November 29th, 2017

Problem Statement

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 of the rectangle.

​Process

In my first attempt at drawing the diagram, I actually got help from my classmates because I had been absent that day, so I got all of my information from my peers. In the beginning, I understood what we were trying to do, and what the equation meant (t's a parabola with it's vertex on the y-axis) but when it came to understanding the rest of the equations and determining the values each point held. After talking and communicating to my group, I was able to understand what we were trying to figure out, and at the same time, understand where we were going with the problem.

For the problem, we needed to plug in various values to different equations in order to figure out the Area of the Rectangle, and the Perimeter of the Rectangle.  Once my group was able to figure out the parabola and it's points, the let me know what they were, and took the time to walk me through the beginning of the problem so that I could best understand it. Our group definitely got lucky in solving the problem, and the hints that were given to us definitely pushed our thinking and helped us solve the problem. With us knowing that our equation was a parabola, that helped us in getting through the rest of the equation.

​Solution

The problem is actually really simple once you go through it and figure it out. It was definitely a new experience for me and I feel like this problem helped me a lot in understanding the equation better. Basically, here is what we did:
  1. We first needed to figure out the Y-Intercept and X-Intercept
  2. Once we figured them out, we needed to plug values into the equation to find different points on the parabola that we had drawn on the board
  3. After this, we tried to divide the space inside the parabola into different rectangles, to try and narrow down the possibilities of our rectangles
  4. From this, we made a XY table, where we would be able to see our values for length and width
  5. To try and find the lengths and widths, we just needed to do x times y
  6. From there on, we realized that the highest value was 2, and to figure out the largest area of them all, we needed to look at all the values inside of 2 and 3
  7. We found the highest value, using the equation ( Area=x(16-x^2), and then plug all the different values of x into this equation
  8. We distributed it out and thats how we got the equation for the area so it would be 16x-x^3. 
  9. We then noticed that in the equation, it started to go up, and then back down, much like a parabola
  10. The point where it's the biggest is at 2.3 as our x and 10.71 as our y
  11. We realized we went one step further using the same equation as the one before, but we took the step to plug in all the numbers between 2.3 and 2.4
  12. The largest area that it had was 2.31 because the next number, 2.32, it started going down
  13. And now, to find the perimeter of the problem, we needed to use the equation 1(x*y)
  14. In the regular (xy) table that we used in the beginning, we plugged in this equation and realized that 0 and 1 had the same perimeter, so we realized we needed to look between those two numbers to find the biggest perimeter the rectangle had
  15. So we tried the numbers .1 through .9 and plugged the numbers into our new equation, which was 2x+32-2x^2
  16. ( The equation is different because we needed to find the value of y, and by doing that, you plug the numbers into the value of y and then distributed out like we did with the area
  17. We found out that .5 gave us the largest perimeter, and to check that it was that number, we tried .51 but it was smaller than our perimeter (32.50)
  18. These were our final numbers that we found:
    1. x = .5
    2. y = 15.75

Group Test / Individual Test

To prepare for our group test, my group and I decided to do the same number, but change the numbers so we could verify that our group understood. The group really helped each other figure out the problem and making sure that we understood everything about how to solve and what each of the equations meant. I do feel like me and Lorena could've made an effort to ask more questions to better understand the problem, because I understood when working on the group test, because I was able to ask questions, but in the individual test, I confused myself and didn't really understand. I really wish me and Lorena could've asked more questions, because they could've helped us to prepare for the individual test.

Evaluation / Reflection

During the unit, I feel like trying to understand the problem and catching up (since I was absent when we first did the problem), was what pushed my thinking. Asking questions was definitely crucial if you didn't understand the problem because if you didn't understand the question, then that's when you would fall back behind and not understand enough for the individual test. I feel like even though I didn't perform very well in the individual test, I  feel like I tried my best in the group test and tried to understand the problem as best as I could. If I would give myself a grade, I would say an B+ because even though I didn't put in my best effort into the individual test, I feel like I understood the problem well enough to explain it here. If I could re-do the problem, I would definitely put more effort into asking the right questions and trying to contribute to the problem by communicating with my group during the test and practices' as well.