A horizontally lying cylinder is fixed by attaching a rib to its side at two points. (The ribs touch the cylinder's side with their upper edges.) What is the radius of the cylinder if the heights of the ribs are 9 and $2 \mathrm{~cm}$, and their distance is $23 \mathrm{~cm} ?$
Solution. Using the notations in the figure, $R$ is the radius of the cylinder.

![](https://cdn.mathpix.com/cropped/2024_05_02_533720a9ed7a3b65782dg-1.jpg?height=279&width=593&top_left_y=218&top_left_x=755)

We apply the Pythagorean theorem to the right triangles:

$$
\begin{aligned}
& R^{2}=(R-9)^{2}+(23-x)^{2} \\
& R^{2}=(R-2)^{2}+x^{2}
\end{aligned}
$$

Subtracting the two equations, we get the equality $0=-14 R+606-46 x$, from which:

$$
R=\frac{303-23 x}{7}
$$

Substituting this into the second equation, we get the equation $7 x^{2}+92 x-1184$, whose solutions are: $x_{1}=8, x_{2}=-21.14$. Clearly, only the first one provides a solution, in which case $R=17$.

Therefore, the radius of the cylinder is $17 \mathrm{~cm}$.