Posted by **Thomas** on Wednesday, November 17, 2010 at 9:57pm.

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!

- Math. HELP! -
**Reiny**, Wednesday, November 17, 2010 at 10:08pm
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 !!!!

- Math. HELP! -
**Thomas**, Wednesday, November 17, 2010 at 10:13pm
Reiny. You. Are. God. Thank you :D

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