Problem

A 20-inch square piece of metal is to be used to make an open-top box by cutting equal-siz...

A 20-inch square piece of metal is to be used to make an open-top box by cutting equal-sized squares from each corner and folding up the sides (as in Example 1). The length, width, and height of the box are each to be less than 14 inches. What size squares should be cut out to produce a box with

(a) volume 550 cubic inches?
(b) largest possible volume?

EXAMPLE 1

A box with no top is to be made from a 22 × 30 inch sheet of cardboard by cutting squares of equal size from each corner and bending up the flaps, as shown in Figure 1. To the nearest hundredth of an inch, what size square should be cut from each corner in order to obtain a box with the largest possible volume, and what is the volume of this box?

SOLUTION

Let x denote the length of the side of the square to be cut from each corner. Then, as we saw in Example 2,

Thus, the equation y = 4x3 − 104x2 + 660x gives the volume y of the box that results from cutting a square of side x from each corner. Since the shortest side of the cardboard is 22 inches, the length x of the side of the cut-out square must be less than 11 (why?).

Each point on the graph of y = 4x3 − 104x2 + 660x (0 < x < 11) in Figure 2 represents one of the possibilities:

The x-coordinate is the size of the square to be cut from each corner;

The y-coordinate is the volume of the resulting box.

The box with the largest volume corresponds to the point with the largest y-coordinate, that is, the highest point in the viewing window. A maximum finder (Figure 3) shows that this point is approximately (4.182, 1233.809). Therefore, a square measuring approximately 4.18 × 4.18 inches should be cut from each corner, producing a box of volume approximately 1233.81 cubic inches.

Figure 1

Figure 2

Figure 3

EXAMPLE 2

A box (with no top) of volume 1000 cubic inches is to be made from a 22 × 30 inch sheet of cardboard by cutting squares of equal size from each corner and folding up the flaps, as shown in Figure 4. If the box must be at least 4 inches high, what size square should be cut from each corner?

SOLUTION Let x denote the length of the side of the square to be cut from each corner. The dashed rectangle in Figure 4 is the bottom of the box. Its length is 30 − 2x as shown in the figure. Similarly, the width of the box will be 22 − 2x, and its height will be x inches. Therefore,

Since the cardboard is 22 inches wide, x must be less than 11 (otherwise, you can’t cut out two squares of length x). Since x is a length, it is positive. So we need only find solutions of the equation between 0 and 11. We graph

y = 4x3 − 104x2 + 660x − 1000

in a window with 0 ≤ x ≤ 11 (Figure 5). A complete graph isn’t needed here, only the x-intercepts (solutions). The one between 2 and 3 is not relevant here because x is the height of the box, which must be at least 4 inches.

GRAPHING EXPLORATION

Use a root finder or a polynomial solver on a calculator or computer to find the solution of the equation between 6 and 7. This is the side x of the square that should be cut from each corner. Round the value of x to two decimal places, and find the dimensions of the resulting box.

Figure 4