Question 864008
Surface Area:  {{{pi*r^2+pi*r^2+h*2*pi*r}}}, using h for length top to bottom.


Material Prices:  Each refers to the material for the top.
Top, x money units per square inch
Curved Side, 2x
Bottom, 3x


Account for Material Cost:  {{{pi*x*r^2+pi*3x*r^2+h*pi*2x*r}}}
{{{pi*x*r^2+pi*3x*r^2+pi*h*2x*r}}}


A function for material cost for the cylinder is wanted, and we have two variables, h and r.  We are assuming x is known, so it is a constant.  We can use the given volume 100*pi to find a formula for h:


{{{h*pi*r^2=100*pi}}}
{{{h=100/r^2}}}
Substitute into the Material Cost expression to make the function.
{{{C=pi*x*r^2+pi*3x*r^2+pi(100/r^2)*2x*r}}}
We do not need to specifically know any particular value for x; it is "the money unit" and for all practical purposes, since the ratios were already given, x=1...
{{{C=pi*r^2+pi*3r^2+pi*2(100/r^2)r}}}
{{{C=pi(r^2+3r^2+2*100/r)}}}
{{{highlight_green(C=pi(4r^2+200/r))}}}
We should be able to look for r-axis intercepts, treating r as the horizontal number line.  ...Maybe not...


{{{highlight(C=pi(4r^2*r+200)/r)}}}
{{{highlight(C=pi(4r^3+200)/r)}}}
-
Either form of this C function would do; this seems to be a Calculus derivative problem.  Find first derivative, and look for a minimum value where the derivative is zero.  
I resorted to Google to look at the graph instead of actually finding and using the derivative.  The minimum looks like it is around r=3 for the minimum cost.