Question 266652
Suppose the volume must be 50in^3, What are the values for r and h that will minimize the amount of sheet metal required to obtain this volume. I know volume is v= (pi)r^2h. I also know the surface area is sa= 2(pi)rh+2(pi)r^2. 
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Eliminate either r or h, then minimize Area wrt the remaining variable.
V = pi*r^2h = 50
h = 50/(pi*r^2)
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A = 2pi*rh + 2pi*r^2 = 2pi(rh + r^2)
A = 2pi(r^2 + 50/pi*r)
A = 2pi*r^2 + 100/r
dA/dr = 4pir - 100/r^2 = 0
pi*r - 25/r^2 = 0
r^3 = 25/pi
r =~ 1.99647 inches
h =~ 3.99295 inches
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It's a tough problem.