# Cone in a cone optimization

• Nov 16th 2009, 08:49 AM
helpplz
Cone in a cone optimization
There is a cone insribed in another cone, one upside down in the other. The two bases are parallel and the vertex of the smaller cone lies at the center of the larger cone's base. The height of the large cone is 12, and the radius of the large cone is 6. What values of radius and height will give the smaller cone the largest possible volume?

^^ Whaaa? How would I even start this? lol. Could anyone walk me through it? xD
• Nov 16th 2009, 09:09 AM
galactus
If you make a drawing you will see similar triangles.

Label the height of the smaller cone h and its radius r.

By the similar triangles we have $\frac{r}{6}=\frac{12-h}{12}$

Solve this for r and sub into the volume of a cone formula.

$\frac{\pi}{3}r^{2}h$

Then, differentiate, set to 0 and solve for h. r will follow.
• Nov 16th 2009, 01:44 PM
helpplz
I don't understand why you put r/6 = 12 - h/h