# Sphere inside a cone problem.

• Jul 4th 2007, 11:57 AM
Patrick_John
Sphere inside a cone problem.
I'm taking a test soon, the next problem is one of which I'll be facing on it, and I would like to ask you guys to help me know how to solve it, as well as which is what I need to study in order to solve alike problems.

Suppose that a sphere of radius 1 is inscribed in a right circular cone with radius r and height h.

(1) Express r in terms of h.

http://img74.imageshack.us/img74/9938/circleos6.jpg

(2) Find the minimum of the volume of such a right circular cone.

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Now the answer for (1) is supposed to be
http://img513.imageshack.us/img513/5246/circle2jf4.jpg

$\displaystyle \sqrt{(h-1)^2 - 1} : 1 = h:r$

$\displaystyle h = r\sqrt{h^2 - 2h}$

$\displaystyle r = \frac{h}{\sqrt{h^2 - 2h}}$

Why is this so?

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And for (2)

$\displaystyle V (h) = \frac{1}{3}\pi r^2h = \frac{1}{3} \pi h \cdot \frac{h^2}{h^2 - 2h} = \frac{1}{3} \pi \cdot \frac{h^2}{h - 2}$

$\displaystyle V'(h) = \frac{1}{3}\pi \cdot \frac{h (h - 4)}{(h - 2)^2}$

$\displaystyle \frac{8\pi}{3}$

Thanks.
• Jul 4th 2007, 12:06 PM
Plato
“Why is this so?”
Because triangle AEO is similar to triangle ADC.
• Jul 7th 2007, 11:12 AM
Patrick_John
Thanks, I think I'm not yet prepared to take this kind of test, I think I'll keep studying and try it in December, but thanks for your help :)