Among the right-angled triangles with a common hypotenuse, which one, when rotated around one of its legs, produces a cone with the maximum volume?
We denote the common hypotenuse by $c$, the leg around which we rotate by $x$, and the other leg by $y$. According to the Pythagorean theorem:

$$
x=\sqrt{c^{2}-y^{2}}
$$

The leg $x$ is the height of the cone, and the leg $y$ is the radius of the base circle. The volume of the cone is

$$
V=\frac{y^{2} \pi \sqrt{c^{2}-y^{2}}}{3}
$$

Since the volume is positive, the expression we obtained (which depends on $y$) has its maximum at the same point where the

$$
9 \pi^{2} V^{2}=y^{4}\left(c^{2}-y^{2}\right)=y^{2} \cdot y^{2} \cdot\left(c^{2}-y^{2}\right)
$$

function does. At the maximum point, it does not change if we multiply the function by 2, but this way we ensure that the sum of the factors is independent of $y$. Thus, the original volume $V$ has its maximum at the same point where the

$$
y^{2} \cdot y^{2} \cdot\left(2 c^{2}-2 y^{2}\right)
$$

function does.

The sum of the three factors is constant: $2 c^{2}$. According to the arithmetic mean-geometric mean inequality, in this case, the product is maximal when the factors are equal:

$$
y^{2}=2 c^{2}-2 y^{2}
$$

from which

$$
y=c \sqrt{\frac{2}{3}}
$$

The other leg is then

$$
x=\frac{c}{\sqrt{3}}
$$

and the maximum volume is

$$
V_{\max }=\frac{2 \pi \sqrt{3} c^{2}}{27}
$$

András Gyene (Bp. V., Eötvös g. III. o. t.)