# Rational function to find minimum surface area of cylinder

I have developed the formula to determine the radius of a cylinder with a fixed volume:

$$f(x) = \sqrt[3]{\dfrac{V}{\pi}}\$$

Substituted into the formula for the surface area of a cylinder, I get the following function. This would give me the minimum surface area of a cylinder for a given volume.

$$S(V) = 2\pi(\sqrt[3]{\dfrac{V}{\pi}})^2+2\pi(2 * \sqrt[3]{\dfrac{V}{\pi}})$$

However, my assignment for class asks for a rational function for this problem. How could I take my existing function and make it rational?

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What's the height of the cylinder? what is $x$? please explain yourself. – nbubis May 6 '12 at 22:24
note that the units in your derivation are wrong: you can't add two numbers with different dimensions. – nbubis May 6 '12 at 22:39

I assume the cylinder in question has $h=r$, so that: $$r=\sqrt[3]{V/\pi}$$ The surface area is then: $$A=2\pi r^2 + 2\pi r h=4\pi r^2=4(\pi V)^{2/3}$$ This of course is not a rational function in $V$ (and never will be), but is a rational function in $r$. Perhaps this is what the assignment means?

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I believe so. It just asks for the minimum surface area for a given volume. – Ethan Turkeltaub May 6 '12 at 23:19