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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.

Please help me solve this!

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  • $\begingroup$ You made few mistakes in your equation. You should have $$ \pi r^2(4/3 r+h)=\pi/12 $$ So $$ 4/3 r^3+r^2 h= 1/12; $$ $\endgroup$ – Alexander Vigodner Oct 13 '14 at 18:23
  • $\begingroup$ Yeah, that was the initial equation, sorry. I'm still missing a restriction, or something to plug in my initial equation. :) $\endgroup$ – belgarion Oct 13 '14 at 18:31
  • $\begingroup$ What does it mean "minimal volume" - minimal with respect to what? $\endgroup$ – Alexander Vigodner Oct 13 '14 at 18:46
  • $\begingroup$ Like, find the smallest/optimal combination of radius and height that amounts to fill the capsule with (12/pi) of volume. The teacher said we usually have a nice formula to derive at the beginning, but we have 2 variables in it (in this case radius and height). We usually have a restriction which gives a numerical value to one of the two variables to replace in the first equation before deriving. Or a relationship between the two shapes, I don't know. $\endgroup$ – belgarion Oct 13 '14 at 19:03
  • $\begingroup$ I am asking again. What does it mean "smallest/optimal combination? Smallest/optimal with respect to what if the volume is fixed? This is the key issue you have here. $\endgroup$ – Alexander Vigodner Oct 13 '14 at 19:17
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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 $G(r)$.

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