Question

A silo is to be made in the form of a cylinder surmounted by a hemisphere. The cost of construction per square foot of surface area is twice as great for the hemisphere as for the cylinder. Determine the dimensions to be use (in terms of V) if the volume is fixed at V and its cost of construction is to be minimum. Neglect the thickness of the silo and waste in construction.

Answer 1

I assume we are not worried about the base of the silo.
So we need the "sleeve" of the silo plus a hemisphere

let the radius be of the hemisphere be r ft and the height of the cylinder part be h ft

surface area (SA) = 2πrh + (1/2)4πr^2
Cost = 1(2πrh) + 2(1/2)4πr^2
= 2πrh + 4πr^2
but πr^2 h + (1/2)(4/3)πr^3 = V , where V is a constant
h = (V - (2/3)πr^3)/(πr^2)
sub that back into Cost = ....
simplify, then take the derivative
set that equal to zero and solve for r
Make sure you treat V as a constant.