An open top bos with a square base is to be constructed from 192 sq ft of material. What should be the dimensions of the box if the volume is to be a maximum? What is the maximum volume?

Answer 1

base --- x by x ft
height -- y ft

area = 4xy + x^2
192 = 4xy + x^2
y = (192 - x^2)/(4x)

V = x^2 y
= x^2 (192-x^2)/(4x)
= 48x - (1/4)x^3
dV/dx = 48 - (3/4)x^2 = 0 for a max of V
(3/4)x^2 = 48
x^2 = 64
x = 8
then y = (192-64)/(32) = 4

max Volume = (8^2)(4) = 256