Calculating the Optimal Dimensions for a cylinder.

[JessicaDay]

I have to calculate the optimal dimensions for a cylinder of this volume, if the amount of materials used to build it is to be kept to a minimum.

The volume of the cylinder is = 498.76cm^3

THIS IS WHAT I HAVE SO FAR,

V= pi R^2 , h= 498.76/pi R^2
S.A =2piR^2 + 2piRh
= 2piR^2 + 2piRv/piR^2
= 2piR^2 + 2piR x 498.76/piR^2
= 2piR^2 + 997.52/r

Now i don't know where to go from here. have i done it right so far?

 

[Fernando Revilla]

= 2piR^2 + 997.52/r

Right. Equivalently $S(R)=2\pi R^2+\dfrac{2V}{R}.$ The minimum of $S(R)$ should be obtained for $R$ such that $S'(R)=0.$ Then,
$$S'(R)=4\pi R-\dfrac{2V}{R^2}=0,\; \dfrac{4\pi R^3-2V}{R^2}=0,\;4\pi R^3-2V=0,\; R=\sqrt[3]{\dfrac{V}{2\pi}}.$$ Could you continue?