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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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    note that the units in your derivation are wrong: you can't add two numbers with different dimensions.2012-05-06

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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.2012-05-06