Question

An open-top rectangular tank with a square base and a volume of 32 ft3 is to be built. What dimensions minimize the amount of material required to build this tank? Show that your result is a minimum.

Answer 1

x = 8 ft

h = 1/2 ft

Explanation:

Let x be side of the base then area of the base is x²

Let h be the height of the tank

Tank volume is 32 ft³ and is 32 = x²*h then h = 32 /x²

Area of base + lateral area = total area (A)

A = x² + 4*x*h ⇒ A = x² + 4*x*(32/x²) A = x² + 128/x

A(x) = x² + 128/x (1)

Taking derivatives on both sides of the equation

A´(x) = 2x - 128/x² A´(x) = 0 2x - 128/x² = 0

(2x² -128) / x² = 0

2x² - 128 = 0

x² =√64

x = 8 ft

The result is minimum since replacing in equation (1) x = 8 we get

A(x) > 0

And

h = 32/x²

h = 1/2 ft