A 1125 cubic feet open-top rectangular tank, with a square base x feet on a side and y feet deep, is being built to catch runoff water.

Question

A 1125 cubic feet open-top rectangular tank, with a square base x feet on a side and y feet deep, is being built to catch runoff water. A realistic cost function might be

C(x) = 6(x²+4xy)+10xy

What dimensions should the tank be to minimize the cost?

Answer 1

V = volume of tank = x²y = 1125. So, y = 1125/x².

C = 6x² + 34xy = 6x² + 34(1125)/x = 6x² + 38250x^-1

dC/dx = 12x - 38250x^-2 = 12x - 38250/x² = (12x³ - 38250) / x²

dC/dx = 0 when 12x³ = 38250. So, x ≈ 14.7 ft and y = 1125/x² ≈ 5.2 ft.