A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank costs $8 per square meter for the base and $4 per square meter for the sides, what is the cost of the least expensive tank?

Question

A tank with a rectangular base and rectangular sides is to be open at the top.  It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters.  If building the tank costs $8 per square meter for the base and $4 per square meter for the sides, what is the cost of the least expensive tank?

Steve · Answer

If we let the length be L and the height be H, then we have

4LH = 36
LH = 9
L = 9/H

The cost
C = 4L*8 + 4*4H*4
C = 32(9/H) + 64H
C = 288/H + 64H

dC/dH = 64 - 288/H^2
dC/dH = 0 when H = √2.5 = 2.12
so L = 4.24

C = 288/2.12 + 64*2.12 = $271.53