Question

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.

Answer 1

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