Optimization- oh how the brain hurts

[griffon]

Q: A solid if formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cm^3. Find the radius of the cylinder that produces the minimum surface area.

OK, I got about halfway through my problem before I got lost.
V of shpere= 4/3(pi)r^3
V of cylinder= (pi)r^2(h)
total V= 2(4/3*pi*r^30) + (pi*r^2*h)

12=pi*r^2(2*4/3*r+h) which turns into 12/pi*r^2=8/3*r+h
so h=(12/pi*r^2) - (8/3*r)

Now I'm lost.

 

[Tide]

And what is the surface area as a function of r and h?

 

[HallsofIvy]

The problem was to minimize the surface area. You have done nothing with the surface area. What is the surface area of a sphere of radius r? What is the area of the curved surface of a cylinder or radius r and length h?