maximum volume

• November 12th 2008, 06:53 AM
ihmth
maximum 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
• November 12th 2008, 07:12 AM
earboth
Quote:

Originally Posted by ihmth
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

I've attached a sketch of the situation.

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

$r^2=h\cdot p$

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

2. The volume of the cone is calculated by:

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

3. Substitute :

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

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

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

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

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

I've got $h = \frac43 a$