putting question in description

Question

A rectangular box with a volume of 320 cubic feet is to be constructed with a square box and top. The cost per square foot for the bottom is 15 cents, for the top is 10 cents, and for the sides is 2.5 cents. What dimensions will minimize the cost? 

Answer 1

Let x is side of bottom and y is height of box Then x²y = 320. Then cost function C(c,y) = 15x² + 10x² + 10xy = 25x² + 10xy; y = 320/x². Then C(x) = 25x² + 3200/x; C'(x) = 50x - 3200/x² = 0; 50x³ - 3200 = 0; x³ = 64;

x = 4 feet. C''(x) = 50 + 6400/x³; C''(4) = 50 + 100 = 150 > 0. So, we have minimum

x = 4 feet; y = 320/16 = 20 feet