A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

Let the side of the square base be x, and the height of the box be h.

The material of the base is x^2, and the material of the four sides is 4xh.

4000 = hx^2

h = 4000/x^2

The total material is

M = x^2 + 4x (4000/x^2) = x^2 + 16000/x

Take the first derivative of M and set equal to 0.

M' = 2x - 16000/x^2 = 0

Multiply by x^2:

2x^3 = 16000

x^3 = 8000

x = 20; h = 10; M = 1200