A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $4 per square foot and the metal for the sides costs $7 per square foot

Question

Find the dimensions that minimize cost if the box has a volume of 50 cubic feet.

Length of base x=



Height of side z=

Answer 1

x²z = 50; Cost C = 8x² + 28xz; z = 50/x²; C = 8x² + 1400/x; C' = 16x - 1400/x² = 0; 16x³ = 1400;p x³ = 175/2;

x = (175/2)^1/3 = 4.44 ft; z = 50/(175/2)^2/3 = 2.54 ft

This is minimum cost because C'' = 16 + + 2800/x³ = 16 + 2800/(175/2) = 48 > 0