Posted by Pat on Thursday, January 31, 2013 at 9:48pm.
The way I read your question:
You have 48 square inches available in total to make the box.
so let the base be x by x inches and the height y inches
total area available = x^2 + 4xy = 48
y = (48 - x^2)/(4x)
volume = x^2 y
= x^2 (48 - x^2)/(4x)
= 12x - x^3/4
d(volume)/dx = 12 - (3/4)x^2
= 0 for a max of volume
(3/4)x^2 = 12
x^2 = 16
x = √16 = 4
if x = 4, then y = (48 - 16)/16 = 2
the box should have a base of 4 inches by 4 inches and a height of 2 inches
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