Optimization problom

Question

A box is to have an open top, vertical sides, a square bottom, and a volume of 108 m^3.

Find the dimensions that minimize the surface area of the box.

Height= ????

Length=????

Answer 1

h=108/x^2, V=108=hx^2

Area=A=bottom + 4 sides= x^2 +4xh

A = x^2 + 432/x

take the derivative, set it =0

A'= 2x-432/x^2 = 0

x=216/x^2

x^3= 216

x = cube root of 216

x=length = 6 meters

h= 108/36= 3 meters high

a 3 m by 6 m by 6 m open topped box minimizes surface area, for a 108 m^3 volume capacity