Question

A container company is tasked to make an open-top rectangular box with a square base. The box must have a volume of 108cm^(3). let the length of the sides of the square base be x cm and the height h cm.
(1) what value of x will produce the minimum outer surface area?
(2) What is the minimum surface area

Answer 1

Volume = x^2 h = 108 = constant

Surface Area = A (x) = x^2 + 4xh
= x^2 + 4x*108/x^2
= x^2 + 432/x

For minimum area,
dA/dx = 0
2x -432/x^2 = 0
x^3 = 216
x = 6 cm (and h = 108/36 = 3)

Minimum area = 36 + (432/6) = 108 cm^2