Find the circular cone of maximum volume inscribed in a sphere of radius a.

I can visualize what the problem is trying to say, but the problem is..I don't know the solution :( Help

-thank you

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- Nov 12th 2008, 06:53 AMihmthmaximum volume
Find the circular cone of maximum volume inscribed in a sphere of radius a.

I can visualize what the problem is trying to say, but the problem is..I don't know the solution :( Help

-thank you - Nov 12th 2008, 07:12 AMearboth
I've attached a sketch of the situation.

1. You are dealing with a right triangle. According to Euklids theorem you have:

$\displaystyle r^2=h\cdot p$

and : $\displaystyle h = 2a - p~\implies~p=2a-h$

2. The volume of the cone is calculated by:

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

3. Substitute :

$\displaystyle V_{cone} = \frac13 \pi \cdot h\cdot p \cdot h$ and

$\displaystyle V_{cone} = \frac13 \pi \cdot h\cdot (2a-h) \cdot h$

4. You have now the volume as a function of h:

$\displaystyle V(h)=-\frac13\pi\cdot h^3 + \frac23 \pi\cdot h^2$

5. Calculate the first derivation of V(h) and solve V'(h) = 0 for h.

I've got $\displaystyle h = \frac43 a$