Optimization: Cone inside a larger cone

• November 16th 2011, 08:14 AM
AXQ4286
Optimization: Cone inside a larger cone
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.
• November 16th 2011, 11:45 AM
earboth
Re: Optimization: Cone inside a larger cone
Quote:

Originally Posted by AXQ4286
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:

$\frac rR = \frac{H-h}{H}$

Solve for r.

3. The volume of the smaller cone is calculated by:

$V_{small\ cone}=\frac13 \cdot \pi \cdot r^2 \cdot h$

Replace r by the result of #2.

4. Use calculus to determine the maximum value of $V_{small\ cone}$