# MATH 12

The total surface area of a square-based open top rectangular box is 12
square units. Find the dimensions of the box such that the volume is the
maximum.

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1. 4 s h + s^2 = 12 so h = (12-s^2)/4s

v = s^2 h

v = s^2 (12-s^2)/4s = (12 s^2 -s^4)/4s

v = 3 s -(1/4) s^3

dv/ds = 0 for max = 3 - (3/4) s^2

(3/4) s^2 = 3

s^2 = 4
s = 2

h = (12 - 4)/8 = 1

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