optimization calculus, cannot find restriction, I am stuck!

A capsule formed by a cylinder and two half spheres on the top and the bottom have a minimal volume of $\pi / 12$. What is the height and radius of the capsule? The volume is $$\pi r^2 (4 \times \pi/3 + h ) = 12 / \pi$$ This is usually derived. I cannot find a restriction to plug in the first formula.

• You made few mistakes in your equation. You should have $$\pi r^2(4/3 r+h)=\pi/12$$ So $$4/3 r^3+r^2 h= 1/12;$$ – Alexander Vigodner Oct 13 '14 at 18:23
The area of the capsule $S(r,h)$ is : $$S(r,h)=2\pi r h +4\pi r^2$$
You have constraint $$4/3r^3+r^2h=1/12;$$ So $rh=1/12/r-4/3 r^2$ and the area as the function of $r$ is $$S(r,h)=2\pi r h +4\pi r^2=2\pi(rh+2r^2)=G(r)=2\pi(1/(12r)-4/3r^2+2r^2)=2\pi(2/3 r^2-1/(12r))$$ Now you have to minimize $G(r)$.