A ball was floating in the water of a lake, forgotten. With the arrival of winter, the lake froze over completely, the ball was removed, leaving an 8 cm deep, 24 cm diameter depression in its wake. What is the radius of the ball in centimeters?
Solution. Assuming the ball is a regular sphere, its vertical cross-section will be a circle. The center of the circle is $O$, the length of the chord $AB$ at a height of $8 \mathrm{~cm}$ is $24 \mathrm{~cm}$, and the midpoint of the chord is $F$. In the right triangle $OFB$, $r^{2}=12^{2}+(r-8)^{2}$. Solving the equation, we find that the radius of the sphere is $r=13 \mathrm{~cm}$.

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