A hemisphere with a radius of $10 \mathrm{~cm}$ protrudes from a horizontal plane (to the annoyance of drivers). A cylinder rolls on the plane and collides with the hemisphere with its side. Let $\alpha$ be the angle between the common tangent plane at the point of contact of the two bodies and the horizontal plane. What is the minimum radius of the cylinder if $\alpha \leq 30^{\circ}$?
Let's construct a plane perpendicular to the horizontal plane through the center $O_{1}$ of the sphere. This plane intersects the cylinder in a circle, whose center we denote by $O_{2}$ and radius by $R$. The foot of the perpendicular from $O_{2}$ to the horizontal plane is denoted by $T$, the common tangent point of the two circles by $E$, and the intersection of the common tangent with the horizontal plane by $M$.

$\mathrm{The} \angle O_{1} O_{2} T = \angle O_{1} M E = \alpha$, since they are acute angles of right triangles.

From the triangle $\triangle O_{1} O_{2} T$, we have $\cos \alpha = \frac{R}{R+10}$, from which

$$
R = \frac{10 \cos \alpha}{1 - \cos \alpha}
$$

If $\alpha = 30^{\circ}$, $\cos \alpha = \frac{\sqrt{3}}{2}$, and $R = 10(3 + 2 \sqrt{3}) \approx 64.64$.

We claim that in the interval $0 < \alpha \leq 30^{\circ}$, this is the smallest value of $R$.

(1) The right-hand side can be transformed as follows:

$$
10 \frac{\cos \alpha}{1 - \cos \alpha} = 10 \frac{\cos \alpha - 1 + 1}{1 - \cos \alpha} = 10 \left( \frac{1}{1 - \cos \alpha} - 1 \right)
$$

If $0 < \alpha < 30^{\circ}$, then as $\alpha$ decreases, $\cos \alpha$ increases, $1 - \cos \alpha$ decreases, and $\frac{1}{1 - \cos \alpha}$ increases (and of course, it will also increase after subtracting 1 and multiplying by 10). The function $R: \alpha \rightarrow \frac{10 \cos \alpha}{1 - \cos \alpha}$ is therefore strictly decreasing if $0 < \alpha < 90^{\circ}$.

Thus, $R$ indeed takes its smallest value at $30^{\circ}$ if $0 < \alpha \leq 30^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_60134dc2cd65767cfa90g-1.jpg?height=385&width=591&top_left_y=1038&top_left_x=775)