Are you sure you've phrased the problem correctly?.

Are you sure it isn't, "find the cylinder of max volume that can be inscribed in a cone of the given dimensions"?.

The surface area of the cone is , where L is the slant height of the cone.

So, I don't think that's what needs maximized.

I may be misunderstanding, though.

It can be shown that the cylinder of max volume that can be inscribed in a cone is 4/9 the volume of the cone.