What is the maximum volume of a cone which fits inside a sphere of volume 1?
1. Draw a sketch. Preferable a sketch of the vertical cross-section.
2. From the volume you can determine the radius of the sphere:
$\displaystyle V_{sphere} = 1 = \frac43 \cdot \pi \cdot a^3$
Solve for a.
3. Let r denote the radius of the base circle of the cone and h it's height.
Then
$\displaystyle V_{cone} = \frac13 \cdot \pi \cdot r^2 \cdot h$
4. According to Euclid's theorem
$\displaystyle (2a-h) h = r^2$
5. Plug in this term into the equation of the volume to get the equation of a function:
$\displaystyle V(h)= \frac13 \cdot \pi \cdot (2a-h) h \cdot h = \frac13 \cdot \pi \cdot (2ah^2-h^3)$
6. Determine the maximum of this function.
Thanks earboth.
I have solved this problem before but thought that some of you might enjoy it.
I really like this problem because it's so simple to state and yet involves quite a range of mathematical techniques form algebra, geometry & calculus.
Anyone care to take it to a conclusion?
Suhada