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.