A box is contructed out of two different types of metal. the metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $4 per square foot. find the dimensions that minimize cost if the box has a volume of 15 cubic feet.

Total cost, C = 8x^2 + 12xz

volume, V = (x^2) * z = 10

C = 8x^2 + 12x*10/x^2

= 8x^2 + 120/x

dC/dx = 16x - 120/x^2 = 0

16x = 120/x^2

x^3 = 120/16

x = 1.957 ft

d^2C/dx^2 = 16 + 240/x^3 = + ve for x = 1.957

so, C is minimum when

x = 1.957 ft z = 2.61 ft

Length of the base x is 1.957 ft and height of side z is 2.61 ft