Calculus
posted by Derek .
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?

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 (192x^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 = (19264)/(32) = 4
max Volume = (8^2)(4) = 256
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