Philippa M. Rhodes

# Final Project

## Part I. , #2

Given a rectangular sheet of cardboard 15 in. by 25 in. If a small square of the same size is cut from each corner and each side folded up along the cuts to form a lidless box.

#### What is the maximum volume of the box? What size(s) of the square would produce a box of Volume equal to 400 cubic inches?

Let's first demonstrate this problem using Geometer's Sketchpad.
We are given a rectangular sheet of cardboard:

Then we will cut a small square of the same size from each corner

and fold each side up along the cuts to form a lidless box.

Since Volume = length * width * height , we can multiply the lengths of the two sides times the height to find the volume. GSP will measure the lenth of each side by simply selecting the side and choosing the 'Measure - Length' tool. We can then use the calculator that GSP provides to compute the volume. So,

### Volume =

By carefully constructing the box, we are able to change the length of the sides of the small squares (the height) all at once, and see how it will change the volume of the box.

We want to find the maximum volume of the box. This can be estimated by simply changing the height to see when the volume is at its highest value.

### Volume =

Click here (for the GSP file) to see how changing the height changes the volume of the box.
Lastly, we want to find the sizes of the box that will produce a volume of 400 cubic inches. Again, we can change the height until the volume is as close to 400 cubic inches as possible. Here, 400.452 was the closest, and the sides are as follows:

### Volume = .

For the second size, the estimation is not as close. So, we the nearest value below 400

### Volume =

and the nearest above 400 in order to obtain an approximation of the sizes.

### Volume =

Next, we can use Algebra Xpresser to 'solve' the problem.

To use Algebra Xpresser, we need a formula for the volume. By looking at the sheet of cardboard,

we see that one side of the box will be 25 minus two times the length of the small square. (Let x= the length of the small square). The other side will be 15 minus two times x, and the height will be x. Thus, the equation for the volume of the box is y = x * (25 - 2x) * (15 - 2x).

We can use the graph to find the box's maximum volume and the sizes when the volume is 400 cubic inches.

#### y = x * (25 - 2x) * (15 - 2x)

By the first graph, the maximum value appears to be 512 cubic inches, but when we zoom in on the top of the parabola,

we get a better estimate, 513.0513 cubic inches. This is very close to 513.032 value that we obtained using GSP.

Next, we can find the values of x when the volume is 400 cubic inches.

When x = 1.53, the length is 25 - 2x = 21.94 inches, and the width is 15 - 2x = 11.94 inches.

When x = 4.8, the length is 15.4 inches, and the width is 5.4 inches.

#### Microsoft Excel

With Excel, start by placing values of x ( length of the side of the small square) in column A. We can start with 0 and increase the values by .25 until x = 7.5 (x can not exceed 7.5 because the width, 15 - 2x, would be negative). In column B, we can compute the volume of the lidless box for each of these values of x. The formula needed is A1 * (25 - 2*A1) ( 15 - 2*A1). Then "fill down" column B.

The first observation is that the box reaches its maximum value when x is between 2.75 and 3.25 inches.

Now, start with 2.75 and increase in increments of .05, and observe that the maximum is when x is between 3 and 3.1 inches.

After narrowing in on the value a couple more times, we can conclude that the maximum volume of the box is 513.051293 cubic inches.

The same technique can be used to find the sizes when the volume is 400 cubic inches.

The closest values for x are 1.525 and 4.7928 inches. Thus, the lengths 21.95 and 15.4144 inches, respectively, and the widths 11.95 and 5.4144 inches.

We can also use Microsoft Excel to graph the data.