What is the maximum volume of a cylinder that can fit in a sphere of a constant radius?

The first question that comes into my mind here is whether any cylinder that touches(at 4 pts) the circumference of the sphere and does not go out of it, has equal volume?

Second, how do i mathematically limit the volume of the cylinder to be less than that of a sphere? Squeeze theorem?

-
Are there spheres of non-constant radius? :) –  Thomas Andrews Nov 27 '12 at 15:31
This is basically the same problem, but with a cone instead of a cylinder. See if you can apply the same ideas. math.stackexchange.com/questions/225718/… –  apnorton Nov 27 '12 at 15:38
You can see that not all such cylinders have equal volume, just by considering the extreme case of when two of the points are very close together. You get either a long thin rod, or a big flat pancake, and the volume of each of these tends to zero as the points become closer together. So the maximum volume must lie somewhere between these two extremes. –  TonyK Nov 27 '12 at 15:58

Let $R$ be the radius of the sphere and let $h$ be the height of the cylinder centered on the center of the sphere. By the Pythagorean theorem, the radius of the cylinder is given by $$r^2 = R^2 - \left(\frac{h}{2}\right)^2.$$

The volume of the cylinder is hence \begin{align} V &= \pi r^2 h\\ &= \pi \left(h R^2 - \frac{h^3}{4}\right). \end{align}

Differentiating with respect to $h$ and equating to $0$ to find extrema gives $$\frac{dV}{dh}=\pi \left(R^2 - \frac{3h^2}{4}\right) = 0\\ \therefore h_0 = \frac{2R}{\sqrt{3}}$$

The second derivative of the volume with respect to $h$ is negative if $h>0$ such that the volume is maximal at $h = h_0$. Substituting gives $$V_{max}=\frac{4 \pi R^3}{3\sqrt{3}}.$$

-
nice solution , just one correction. When applying Pythagorean theorem, the exact way would be $r^2+(h/2)^2=R^2$ where $r$ is the radius of small cylinder and $R$ is the radius of big cylinder and $h$ is the height of cylinder. –  ashmantak Mar 28 '14 at 10:26
Thanks for the catch. That is now fixed. –  Till Hoffmann Mar 28 '14 at 11:17

Hint: In the context of a calculus course, I think you are first expected to argue informally that such a maximal cylinder must have axis that goes through the center of the circle, and that without loss of generality that axis is the $z$-axis.

So now suppose that the cylinder meets the $x$-$y$ plane in a circle of radius $t$. Find the height of the cylinder in terms of $t$, and hence the volume. Now use the ordinary tools to maximize.

-