We cut a circular piece of paper into $n$ identical sectors, and then shape each of the resulting sectors into a cone. For what value of $n$ will the total volume of the cones determined by the cone-shaped sectors be maximized?
Let the radius of the circle be $R=1$, the central angle of a sector $\frac{360^{\circ}}{n}$, and the arc length $i=\frac{2 \pi}{n}$.

The radius of the base circle of the cone is $r$. From the equation $2 r \pi=\frac{2 \pi}{n}$,

$$
r=\frac{1}{n}
$$

The height of the cone can be calculated using the Pythagorean theorem:

$$
m=\sqrt{1-\left(\frac{1}{n}\right)^{2}}=\frac{1}{n} \sqrt{n^{2}-1}
$$

The total volume of the cones is:

$$
V=\frac{n\left(\frac{1}{n}\right)^{2} \pi \frac{1}{n} \sqrt{n^{2}-1}}{3}=\frac{\pi}{3} \frac{\sqrt{n^{2}-1}}{n^{2}}
$$

The question is, when will this sum be maximal.

The expression $\frac{\sqrt{n^{2}-1}}{n^{2}}=\frac{1}{n} \sqrt{1-\frac{1}{n^{2}}}$ "visibly" decreases as $n$ increases (and $n \geq 2$, since we cut the paper into at least two sectors):

![](https://cdn.mathpix.com/cropped/2024_05_02_3dc67c955b7003563c69g-1.jpg?height=180&width=772&top_left_y=948&top_left_x=183)

The graph of the function $y=\frac{\sqrt{x^{2}-1}}{x^{2}}$ is shown in Figure 3, created using a computer.

It is expected, therefore, that the maximum will be obtained when $n=2$. To prove this, we need to show that

$$
\frac{\sqrt{3}}{4}=\frac{\sqrt{2^{2}-1}}{2^{2}}>\frac{\sqrt{n^{2}-1}}{n^{2}}, \quad \text { if } n>2
$$

Squaring both sides (with $n^{2}-1>0$) and rearranging,

$$
3 n^{4}-16 n^{3}-16>0
$$

This is a quadratic expression in $n^{2}$. Let's complete the square:

$$
9 n^{4}-48 n^{2}-48=\left(3 n^{2}-8\right)^{2}-16>0
$$

From this, $\left(3 n^{2}-8\right)^{2}>16$. If $n>2$, then $3 n^{2}>8$, and

$$
\left(3 n^{2}-8\right)^{2}>\left(3 \cdot 2^{2}-8\right)^{2}=4^{2}=16
$$

This proves our statement. The sum of the volumes is maximal when the circle is divided into 2 sectors.
![](https://cdn.mathpix.com/cropped/2024_05_02_3dc67c955b7003563c69g-1.jpg?height=772&width=420&top_left_y=1874&top_left_x=836)

![](https://cdn.mathpix.com/cropped/2024_05_02_3dc67c955b7003563c69g-2.jpg?height=398&width=1114&top_left_y=129&top_left_x=492)