First, what is the relationship between the height of the cylinder and the radius of the cylinder?

If you make a sketch, you'll see that for a certain radius r, the cone would have a height of l = (r/6)*20, using similar triangles.

From here, the height of the cylinder will become h = 20 - (20r)/6.

Then, the formula for the volume of a cylinder is

Using the equation you got, you now get:

Simplifying;

For the second part, are you sure that's the question? Because r and h would vary, depending on the volume. If the question was in fact, "Find the value of r and h, provided that the volume of the cylinder is maximised." there there would be a solution.