You are in charge of a tank to be used for water storage. Your boss has told you that he wants a cylindrical tank with a minimum capacity of 300,000 cubic feet. Your boss wants to spend the least possible amount in painting the tank. Design a tank to meet this objective.

Answer 1

Let y be the height. The diameter D must satisfy
(pi/4) D^2 y = 3*10^5 ft^3

The area to be painted (including the top) is
A = pi [D^2/4 + D y]
Substitute using the first equation to eliminate y.
A = pi [D^2/4 + 3*10^5 (4/pi)(1/D)]

Set dA/dD = 0 and solve for D

D/2 - (12*10^5/pi)/D^2 = 0
D^3 = 24*10^5/pi
D = 92.6 ft