Optimization Question

Question

An open top box with a square base and rectangular sides is to have a volume of 9 cubic feet. The cost to make the base is 2 square feet and the cost of the material to make the sides is 3 square feet. Find the dimensions of the box that will minimize the cost.

Answer 1

Let x: side length of square base ; h: height of box

Constraint: V = x²h = 9 so h = 9/x²

Lateral area = 4xh so cost of sides = 12xh

Base area = x² so cost of base = 2x²

Cost, C = 2x² + 12xh

C(x) = 2x² + 12x(9/x²) = 2x² + 108/x

C^'(x) = 4x - 108/x² = 0

4x³ - 108 = 0

x³ = 27 ; x = 3 ; h = 1