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? 
Please help, thanks!

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. 
