A rectangular box is to have a square base and a volume of 20 ft3. If the material for the base costs $0.35 per square foot, the material for the sides costs $0.10 per square foot, and the material for the top costs $0.15 per square foot, determine the dimensions of the box that can be constructed at minimum cost.

Surface Area = 2 (lw + lh + wh)

but l = w

Surface area = 2 (l^2 + lh + lh)

Cost of the box = 0.35l^2 + 4 (0.10) lh + 0.15l^2 = 0.5l^2 + 0.4lh

Volume = lwh = 20ft^3

l^2h = 20

h = 20/l^2

Cost = 0.5l^2 + 0.4l (20/l^2) = 0.5l^2 + 8/l

For minimum cost

l - 8/l^2 = 0

l^3 = 8

l = 2

Therefore, for minimum cost the dimensions will be 2ft by 2ft by 5ft