# Calculus (optimization problem)

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A cyclinderical tank with no top is to be built from stainless steel with a copper bottom. The tank is to have a volume of 5ð m^3. if the price of copper is five times the price of stainless steel, what should be the dimensions of the tank so that the cost is a minimum?

• Calculus (optimization problem) -

Volume = V = pi r^2 h = constant
so pi r^2 = V/h
and r =(V/[pi h])^.5
and

cost = 5 pi r^2 + 2 pi r h
cost = 5 V/h + 2 pi r h
cost = 5 V/h + 2 pi (V/pi)^.5 h^.5
d cost/dh = -5 V/h^2 + 2 pi (V/pi)^.5 (.5)(h^-.5)
0 when
5 V/h^2 = pi^.5 V^.5 h^-.5

h^1.5 = 5 V^.5/pi^.5
h^3 = 25 V/pi

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