Sarah Donaldson

EMAT 6680

11/4/06

Assignment 12 Write-Up

Problem 6: Maximizing the Volume of a Lidless Box

Before beginning work on this assignment, I had never used Excel before. Being pretty clueless about how to use spreadsheets in general, I have chosen a relatively simple problem for which to do a Write-up for this assignment. Though not particularly difficult to someone familiar with Excel (and spreadsheets in general), this problem challenged me and allowed me to practice my new (very basic) skills with this tool.

Here is the problem: Given a rectangular piece of cardboard with dimensions 5 units x 8 units, cut out a square from each corner so that, when the sides are folded up, a lidless box of maximum volume is formed.

The formula for the volume of a rectangular prism is V = lwh. In this case, the volume is:

V = (8 – 2x)(5 - 2x)x

In order to examine how various values for x affect the volume, we can use the following Excel spreadsheet and accompanying graph:

x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
/ Volume
0
3.744
6.992
9.768
12.096
14
15.504
16.632
17.408
17.856
18
17.864
17.472
16.848
16.016
15
13.824
12.512
11.088
9.576
8
6.384
4.752
3.128
1.536
-1.33227E-14
/

This table of numbers and its graphical representation suggest that the maximum volume of 18 cubic units occurs when the length of the cut-out square is 1 unit. It is worth noting also that there are limitations on the value of x. It must be greater than 0 (in order to have a volume greater than 0), and must be less than 2.5 units. This upper limit is because the width of the original rectangle is 5 units, so there is no room to cut out 2.5 units twice. These limitations are seen in both the table and the graph. At x = 0 and x = 2.5, the volume is 0 (for all practical purposes, -1.33227E-14 is the same as 0).

Ideas for classroom use:

This activity could be used in a course as basic as 6th or 7th grade math—any course in which students are familiar with finding the volume of rectangular prisms. The activity could also be used in a high school classroom (algebra or geometry) to explore the date more deeply. In an algebra course, the maximization could be analyzed from an algebraic standpoint. In a geometry course, the focus could be on volume. This activity could even be extended to compare volume with surface area and discuss the pros and cons of a number of combinations of these values.

In any case (at any level) a good introduction to the activity is to have students begin with an actual rectangular piece of cardboard. Students could be put into groups, and each member of each group cut out a different size square from each corner of his/her cardboard. Group members could then compare the volumes and propose a conjecture about the maximum volume. It would also be good to discuss the smallest and largest possible squares that could be cut out (to determine the boundaries/domain of x).

Since it’s not practical for students to cut out miles and miles of cardboard, the next step in this lesson should be to create a table of values (for which Excel is an excellent tool). Algebra students could be expected to come up with the formula for volume in terms of x—middle schoolers may need more assistance. However, all students should be able to interpret the table of values and accompanying graph. Both are good models that show the maximum volume quite clearly. Do these results fit with the volumes of the actual cardboard boxes? Did you expect these results? Why does x have to be between 0 and 2.5? These are all good questions to pose to students at any level.