# Thread: Maximum volume of cone in sphere

1. ## Maximum volume of cone in sphere

What is the maximum volume of a cone which fits inside a sphere of volume 1?

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:

$V_{sphere} = 1 = \frac43 \cdot \pi \cdot a^3$

Solve for a.

3. Let r denote the radius of the base circle of the cone and h it's height.

Then

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

4. According to Euclid's theorem

$(2a-h) h = r^2$

5. Plug in this term into the equation of the volume to get the equation of a function:

$V(h)= \frac13 \cdot \pi \cdot (2a-h) h \cdot h = \frac13 \cdot \pi \cdot (2ah^2-h^3)$

6. Determine the maximum of this function.

3. 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?