Minimum surface area.

Question

Minimum surface area. Open top square based rectangular box that have volume of 32(ft 3). What dimensions will minimize surface area? What is the minimum surface area? step by step result. Thanks

Answer 1

V=32 = hx² where V=volume,

h=height of the box and x² is the area of the bottom of the box

h=32/x²

The surface area A = 4 sides + bottom = 4hx + x² = 4(32/x²)x + x² = 128/x + x²

Take the derivative of A and set it equal to zero, then solve for x

A'(x) = -128x^-2 + 2x = 0 divide by 2 and multiply by x²

-64 + x³ = 0 x³=64 x=cube root of 64 = 4 feet

h=32/x² = 32/(4)² = 32/16 = 2 feet

The box is 2 x 4 x 4 feet

the minimum surface area = 4(2x4) + 4² = 32 + 16 = 48 square feet