Hint: Draw a picture then find a relationship between the variables using similar triangles.
The question:
Find the dimensions of the right circular cylinder of greatest volume that can be inscribed in a right circular cone of radius R and height H.
I called the radius of cylinder r and the height of the cylinder h. All I've really been able to do is make the relationship that
(H-h)/r = H/R
Apart from that, I am 100% lost. If this had even one number I think I could solve this. Any answers, suggestions, or hints?