You are good up to here.

No, the volume of the cylinder is not equal to the volume of a sphere. You have already used "fits inside a sphere of radius 1" to deduce that . Differentiate and set the derivative equal to 0 to find the r that gives maximum volume.Pi(r^2)(sqrt(1^2 - r^2)) = 4/3(Pi)(r^3)

Now I just need to solve for r. Is this correct? Where does the differentiation come into it?