Dimensions?

Question

Find the dimensions of a rectangular box with square base and open top with volume 38000 cubic centimeters which minimizes the amount of materials used.

Answer 1

Let x denote the length (in cm.) of a side of the square base.
Let y denote the height (in cm.) of the box.

Volume V = length × width × height = x²y

and we are told that the volume is 38000 so

x²y = 38000

The surface area A of the box is the area of the base, plus the area of the four vertical sides. The area of the base is x², and the area of each vertical side is xy. So

A = x² + 4xy

Solve the volume equation for y in terms of x; then use that to substitute for y in the area equation. This will give you A as a function of x. Then minimize A.