A right circular cone has a base radius 5 and altitude 12. A cylinder is to be inscribed in the cone so that the axis of the cylinder coincides with the axis of the cone. Given that the radius of the cylinder must be between 2 and 4 inclusive, find the value of that radius for which the lateral surface area of the cylinder in minimum. Justify your answer. (Note: The lateral surface of a cylinder does NOT include the bases.)
Again I'm not good with Calculus word problems. Am I suppose to find the derivative of the lateral surface area of the cylinder? How do I handle the variables r and altitude? Help please. I really tried to get it, but I'm not use to them.