A ball rolled into the pool and floated in the water. Its highest point was $2 \mathrm{~cm}$ above the water surface. The diameter of the circle marked by the water level on the ball's surface was $8 \mathrm{~cm}$.

Determine the diameter of Julince's ball.

(L. Hozová)

Hint. What is the relationship between the radius of the ball, the radius of the circle marked by the water level, and the distance from the center of the ball to the water surface?
The following figure shows a cross-section of a ball passing through its center (point $S$) and perpendicular to the water surface (line $A B$). Point $C$ is the foot of the perpendicular from point $S$ to the water surface, and point $D$ is the highest point of the ball above the water.

![](https://cdn.mathpix.com/cropped/2024_04_17_5752a8a8cd9a6cffa1f9g-4.jpg?height=546&width=512&top_left_y=184&top_left_x=772)

From the problem, we know that $|A C|=4 \mathrm{~cm}$ and $|C D|=2 \mathrm{~cm}$. The radius of the ball $|S A|=|S D|$ is denoted as $r$. According to the Pythagorean theorem in the right triangle $A C S$, we get:

$$
\begin{aligned}
r^{2} & =4^{2}+(r-2)^{2}, \\
r^{2} & =16+r^{2}-4 r+4, \\
4 r & =20, \\
r & =5 .
\end{aligned}
$$

Julin's ball had a diameter of $10 \mathrm{~cm}$.