Question

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of its base is twice the width. Material for the base costs 14 dollars per square meter. Material for the sides costs 8 dollars per square meter. Find the cost of materials for the cheapest such container.

Answer 1

Volume: V = w L h <=> 10 = 2 w² h
-> h = 5/ w²
Surface Area: bottom + sides
SA = 2w² + 2 wh + 2 ( 2w h)
Replace h = 5/w²:
= 2( w² + 5/w + 10/w)
= 2 ( w² + 15/w )

Cost C = 2 ( 14 w² + 120/w )
Minimize Cost, C' = 0
C' = ....
Thus C (w) = ...
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