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:
$\displaystyle r^2=h\cdot p$
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 $\displaystyle h = \frac43 a$