Question 858609
Let h = height, r = radius.
Let A = surface area
Let v=volume


{{{A=2*pi*r^2*h+pi*r^2}}}.

{{{v=h*pi*r^2=20pi}}}.
Solve this for h:
{{{h=20/(r^2)}}}, and substituting into the A equation,
{{{A=2*pi*r(20/(r^2))+pi*r^2}}}
{{{A=40*pi/r+pi*r^2}}},which you will use for applying the costs of the two different parts.


Now, applying the cost to those two area parts, cost as a function of r, C(r) becomes:
{{{C(r)=(0.32)40*pi/r+(0.80)pi*r^2}}}
{{{highlight_green(C(r)=12.8*pi/r+(0.80)pi*r^2)}}}.
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You want to minimize the cost.  Find the derivative of C with regard to r, set equal to zero, and solve this for r.  You then use this to find your value of h.


You might also want to check your result using a graphing calculator for the cost equation without use of Calculus.