I'm having trouble on how to do this problem or where to go on this... I know what the picture would look like and that's about it. Any help would be really appreciated! Thank you.
A right circular cone has base radius 5 and altitude 12. A cylinder is to be inscribed in the cone so that the axis 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 is minimum. Justify your answer and note that the lateral surface of a cylinder does NOT include the bases.