Question

A cylinder is to be made of circular cross-section with a specified volume. Prove that if the surface area is to be a minimum, then the height of the cylinder must be equal to the diameter of the cross-section of the cylinder.
Maybe it's the wording, but I have not been able to crack this one for the past half-hour!

Answer 1

let the radius be r and the height be h
let the volume be V, where V is a constant
πr^2h = V
h = V/(πr^2)

Area = 2 circles + rectangle
= 2πr^2 +2πrh
= 2πr^2 + 2πr(V/πr^2)
= 2πr^2 + 2V/r
d(Area)/dr = 4πr - 2V/r^2
= 0 for a max/min of area
4πr = 2V/r^2
r^3 = V/(2π) = πr^2h/2π
r = h/2
or
2r = h
diameter = height !!!!

Answer 2

Reiny. You. Are. God. Thank you :D