Solve the applied problem.Suppose that the open-top box being made from a sheet of cardboa...

Solve the applied problem.

Suppose that the open-top box being made from a sheet of cardboard in Example 1 may have any height, but is required to have at least one of its dimensions greater than 18 inches. What size square should be cut from each corner?

EXAMPLE 1

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 1. 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 1 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 2). 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 1

Figure 2