A vertical-axis, open-top, straight circular cylinder-shaped container filled with water to the brim is tilted by $30^{\circ}$. How much water flows out of the container if the radius of the cylinder is $10 \mathrm{~cm}$ and its height is $25 \mathrm{~cm}$? Construct the angle at which the container must be tilted so that half of the originally contained water flows out.
On the first figure, we drew the front view of the overturned cylinder. The image of the cylinder's base is the segment $AB$, and the image of the horizontal water surface after the water has flowed out is $BD$. If we cut the cylinder with a plane perpendicular to its axis and containing the diameter $DC$, the volume of the thus cut cylinder is twice the volume of the water that has flowed out, since the two bodies, whose images on the drawing are $ABD$ and $CDB$, are centrally symmetric with respect to the midpoint of the diameter $DB$. From the right triangle $ABD$, $\operatorname{tg} 30^{\circ}=\frac{AD}{AB}$, from which $AD=\frac{20}{\sqrt{3}}$, so the volume of the water that has flowed out is

$$
\frac{1}{2} \cdot 10^{2} \pi \frac{20}{\sqrt{3}}=\frac{10^{3} \pi}{\sqrt{3}} \approx 1813.8 \mathrm{~cm}^{3}
$$

The water will flow out halfway when the container is further tilted from the position shown in the first figure until the image of the surface $FB$ becomes horizontal (second figure). In this case, the angles marked as $\alpha$ on the figure are equal. The angle $\alpha$ is constructed by drawing the diagonal of a $20 \times 25$ rectangle, and $\alpha$ will be the angle that the diagonal of the rectangle makes with the shorter side.

Note. The value of $\alpha$ is approximately $51.3^{\circ}$. If we were to ask how much water would flow out if the container were tilted by $60^{\circ}$, we could not answer using the described method, because in this case the volume of the water that flows out is more than half the volume of the cylinder. In such a case, we proceed as follows: by multiplying the height of the cylinder by a suitable positive number, we apply the above method to a "longer" cylinder.

Lóke Tamás (Pápa, Türr István Gymn., 10th grade) and Papp Dávid (Budapest, Szent István Gymn., 10th grade)
![](https://cdn.mathpix.com/cropped/2024_05_02_d5dccfdc60c9d05a52a4g-1.jpg?height=946&width=480&top_left_y=1002&top_left_x=817)