Surface area of cylinders

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• May 3rd 2011, 09:01 AM
nikki1234
Surface area of cylinders
Hi

This may seem like a very simple problem.

If you are given the surface area of a sphere how do you find the radius and height. For example if the surface area of a cylinder including both ends in 60^2 how would you find the height and radius.

Any help would be appreciated
• May 3rd 2011, 09:19 AM
TheEmptySet
Quote:

Originally Posted by nikki1234
Hi

This may seem like a very simple problem.

If you are given the surface area of a sphere how do you find the radius and height. For example if the surface area of a cylinder including both ends in 60^2 how would you find the height and radius.

Any help would be appreciated

You say sphere once and cylinder twice so I will guess you mean cylinder.

The area of the top and bottom are $\pi r^2+\pi r^2=2\pi r^2$

And the area of the side is $2 \pi r h$

So the total surface area is

$2 \pi r^2 +2 \pi r h =60 \iff \pi r^2 + \pi r h=30$

Without more information this problem has an infinite number of solutions
• May 3rd 2011, 09:29 AM
nikki1234
Hi

Its actually a differentiation problem

the question states if the area of a cylinder including both ends is 60 cm what is the largest possible volume?

Hope that helps in helping me!
• May 3rd 2011, 10:08 AM
TheEmptySet
Quote:

Originally Posted by nikki1234
Hi

Its actually a differentiation problem

the question states if the area of a cylinder including both ends is 60 cm what is the largest possible volume?

Hope that helps in helping me!

Well you know the volume of a cylinder is

$V=\pi r^2 h$

and from above you can solve for h to get

$h =\frac{30-\pi r^2}{\pi r}$

If you put this into the volume formula you get

$V=\pi r^2\left( \frac{30-\pi r^2}{\pi r}\right)=-\pi r^3+30r$

Now you need to maximize the volume.

Since this is a cubic if you want an analytic solution you will need to use calculus and the derivative. If you do not know calculus plot the function on a graphing device and find the maximum volume.
• May 3rd 2011, 10:28 AM
nikki1234
Thank you that was a great help