A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h=1/3H.
A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h=1/3H.
1. Draw a sketch!
2. Let R denote the radius of the larger cone and r the radius of the smaller cone.
Use proportions:
Solve for r.
3. The volume of the smaller cone is calculated by: