Before completing the final assignment problem, I will be doing another exploration from assignment 12. As I said in that assignment, I am not too familiar with the capabilities of spreadsheets, so I took this additional opportunity to explore more into this form of technology.
For this additional exploration I looked into the problem where we have a 5ft by 8ft sheet and we want to maximize the volume by creating an open top box. To do this we must remove a square of the same edge length from each corner of the sheet. To visualize this problem I first made a GSP sketch of what we are starting with and what we are cutting from each corner. Below is this sketch, where the yellow squares illustrate what we will be removing from the sheet. Additionally, these squares of length x will form our height of the open top box. When the squares are removed, and the box is formed, we will have a box with length (5-2x), width (8-2x), and height (x).
Using the above information we can create an Excel spreadsheet which will show how various values of x affect the volume of the open top box we are creating. Thinking about the volume before starting the spreadsheet, we can see that there are limitations on the value of x. If x is longer than half the length (5ft), our open top box will 'disappear.' In addition, since x is a length, it cannot be negative. Therefore, the value of x that will yield maximum volume must be between the values of 0 and 2.5 ft. Below is a snapshot of my Excel spreadsheet as I began to explore the volume of the open top box for various values of x, our height.
Knowing that volume is a cubic function, I decided to find the value of x that yields the maximum volume of the open top box by plotting the value of x with the volume values on a scatter plot. From there, I found the cubic function that best fits this data. Below are the snapshots from this part of the exploration.
From the above scatter plots, we find that the appropriate cubic function that exactly fits (squared error value = 1) the data is V= 4x^3-26x^2+40x+7E-13. Looking at these images, and drawing from our mathematical knowledge, we can see that there is a clear maximum volume value. To find this value, I added the use of my graphing calculator. In the graph of my calculator I entered the volume function of the data, I then used my Calc tools to find the maximum volume value, and the x-value that yields it. After doing this process, I found that the x- value of 1 ft (our height) will yield the maximum volume value for this open top box, 18 cubic feet. Below is my Excel file for any of your further investigations.
Open Top Box
Final Assignment Problem: Finding Area of Shaded Region
First, I constructed the image of a square with four quarter circles centered at each vertex of the square, and the shaded region. My steps to constructing this image is as follows:
1) Construct a square (using GPS tool)
2) Using the side length as the radius and each vertex of the square as the centers, construct 4 circles.
3) Construct the arc (quarter circle) for each vertex, and hide the remaining portions of the 4 circles.
4) Highlight the intersections of the intended shaded region, and construct interior of quadrilateral. Create 4 arcs and their interiors to complete the shading.
In addition to the images above, I have included my GSP file that contains my construction.
Shaded Region of Square
Now, we can begin solving for the area of the shaded region. We are to solve for the area such that the side of the square is of length s, thus the area will be in terms of s as well.
Let's start by finding the area of the interior rectangle formed by the four vertices of the shaded quadrilateral. I have placed a few images from the process below. But the mathematical work and explanation can be found at the bottom of the page, at the link GSP file. This file contains my work and reasoning.