# Thread: Optimization problem

1. ## Optimization problem

Dear forum members,

in a problem I was asked to show that when the surface area of a fixed cylinder was at its minimum, the height of the cylinder would be equal to the diameter of the cross section of the cylinder.

I formed the equation for the surface area of the cylinder, and the differentiated it with respect to r(not sure why I choose r, though. Perhaps because it was the only factor that would influence both the area of the "bottoms" of the cylinder as well as the "body".).

A(r)=2pi*r^2 + 2 pi*r*h

A'(r)=4pi*r+2pi*h

solving for the stationary points of the derivative gives 2r=-h .
The result shows basically what I wanted to show, but the minus is probably not supposed to be there.

However, I noticed, that when I expressed h in terms of the volume and the radius

h=V/pi*r^2

and then plugged that into the equation above and differentiated, the result was 2r=h.

Could someone explain to me why these two different ways of doing give a different result?

All help is appreciated.

2. Originally Posted by Coach
Dear forum members,

in a problem I was asked to show that when the surface area of a fixed cylinder was at its minimum, the height of the cylinder would be equal to the diameter of the cross section of the cylinder.

I formed the equation for the surface area of the cylinder, and the differentiated it with respect to r(not sure why I choose r, though. Perhaps because it was the only factor that would influence both the area of the "bottoms" of the cylinder as well as the "body".).

A(r)=2pi*r^2 + 2 pi*r*h

A'(r)=4pi*r+2pi*h ... this derivative is incorrect, you cannot treat h as a constant.

solving for the stationary points of the derivative gives 2r=-h .
The result shows basically what I wanted to show, but the minus is probably not supposed to be there.

However, I noticed, that when I expressed h in terms of the volume and the radius

h=V/pi*r^2

and then plugged that into the equation above and differentiated, the result was 2r=h.

... this is correct because V is a constant, and the variation of h is related to the variation of r.

Could someone explain to me why these two different ways of doing give a different result?
.

3. Thank you so much for your reply!

But why can't I treat h as a constant?(I'm sorry, this is a reall stupid question, but I don't understand why is it incorrect to treat h as a constant)?

4. Originally Posted by Coach
Thank you so much for your reply!

But why can't I treat h as a constant?(I'm sorry, this is a reall stupid question, but I don't understand why is it incorrect to treat h as a constant)?
$V = \pi r^2 h$

if $V$ is constant and $r$ changed, would $h$ stay the same?