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:
$\displaystyle \frac rR = \frac{H-h}{H}$
Solve for r.
3. The volume of the smaller cone is calculated by:
$\displaystyle V_{small\ cone}=\frac13 \cdot \pi \cdot r^2 \cdot h$
Replace r by the result of #2.
4. Use calculus to determine the maximum value of $\displaystyle V_{small\ cone}$