help in math?

Question

A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.

Answer 1

Start drawing pictures.

A cross section of the cone looks a lot like an isocecles triangle. The base is 2r, and the perpendicular bisector of the base has length h.

A cross section of the cylinder is a rectangle, one of whose sides is on the base, and the other of whose sides cuts the triangle so as to make a smaller, similar triangle.

Let 2s be the sidelength parallel to the triangle's base -- i.e., s is the radius of the cylinder. Let c be the height of the cylinder. You're maximizing V = pi * c * s^2.

Now go back to the picture. By the similarity of the triangles, you'll find that r/h = s/(h-c). That gives you s as a function of c, and hence V as a (cubic polynomial) function of c, where c is of course constrained to be in [0,h]

I presume you can finish from there?