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.
Draw the side view, being an isosceles triangle with a rectangle inside it.
Note that the rectangle, in touching the sides of the triangle, cuts off smaller triangles which must be similar to half of the original triangle.
Note that, whatever the value of the cylinder's radius "r" is, the base of the lower small triangle (let's look at the one on the right of the rectangle) is 5 - r. Also, the height "h" of this smaller triangle is related, by similarity, to the height of the original triangle by:
. . . . .
Solve this relation to get "h" in terms only of "r", noting that the height of the smaller triangle is also the height of the cylinder. Then try to create a formula for the surface area of the "sides" of the cylinder in terms only of "r", and optimize. Remember to check the endpoints of the given interval.