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Unread 13-03-2005, 20:45
sanddrag sanddrag is offline
On to my 16th year in FRC
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Proving smallest surface area of cylinder is when h=2r

I need to prove that surface area A of a cylinder (of volume V) is smallest when the height, h equals twice the radius, r.

We have two equations

1. V=(pi)(r^2)(h)
2. A=2(pi)(r^2) + 2(pi)(r)(h)

Here's what I think I need to do in order to prove the smallest A occurs when h=2r. I think I need to take equation 1 and solve for h which comes out to h=V/(pi)(r^2). Then I need to substitue that in for h in equation 2. This would make equation 2 become:

A=2(pi)(r^2) + 2(pi)(r)[V/(pi)(r^2)]

Now, I know this can be simplified, but I'm not sure if I'm doing it right. I think it would go to:

A=2(pi)(r^2) + 2V/r

Is this right? If so, what do I need to do now? I know I have to take the derivative to find the absolute minium. This is where I'm having trouble. Can someone please help? Thanks.
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Last edited by sanddrag : 13-03-2005 at 20:48.
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