Find the dimensions of the rectangular box with a square base and no top that has volume 2...

Find the dimensions of the rectangular box with a square base and no top that has volume 20,000 cubic centimeters and the smallest possible surface area. [Hint: See Example 1.]

EXAMPLE 1

Set up this problem: A rectangular box with a square base and no top is to have a volume of 20,000 cubic centimeters. If the surface area of the box is 4000 square centimeters, what are its dimensions?

SOLUTION

Read: We must find the length, width, and height of the box. Label: Let x denote the length. Since the base is square, the length and width are the same. Let h denote the height, as in Figure 1. Translate: Recall that the volume of a box is given by the product length × width × height and that the surface area is the sum of the area of the base and the area of the four sides of the box. Then we have these translations:

Consolidate: We have two equations in two variables, so we solve the first equation for h

and substitute this result in the second equation:

The solution of this last equation will provide the solution of the problem.

Figure 1