\section*{Problem 2 - 311022}

A sphere \(K\) is drilled through cylindrically such that the axis of the drill passes through the center of the sphere. After this, a remaining body \(C\) is left.

This ring-shaped body \(C\) has two circular edges; the only given measurement is the length \(h\) of a line segment parallel to the drill axis, connecting a point on one edge with a point on the other edge.

Andrea claims that the volume \(V(C)\) of \(C\) can be determined solely from \(h\).

Birgit, on the other hand, argues that since the given value \(h\) can be achieved from different-sized spheres (radius \(R\)) by choosing a radius \(r\) of the cylindrical drill hole that matches \(R\), different values of \(V(C)\) should be possible for this \(h\); thus, if neither \(R\) nor \(r\) is known, \(V(C)\) cannot be determined solely from \(h\).

Who is right?

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1594.jpg?height=257&width=282&top_left_y=2370&top_left_x=887)

Hint: For the volume \(V(K), V(A), V(Z)\) of a sphere \(K\) (radius \(R\)), a spherical cap \(A\) (height \(p\) see figure), and a cylinder \(Z\) (base circle radius \(r\), height \(h\)), the formulas are

\[
V(K)=\frac{4}{3} \pi R^{3} \quad ; \quad V(A)=\pi R p^{2}-\frac{1}{3} \pi p^{3} \quad ; \quad V(Z)=\pi r^{2} h
\]
}
The claim that the volume \(V(C)\) of a residual body \(C\) can be determined solely with \(h\) is precisely refuted when there exists a residual body \(\bar{C}\) with the same height \(h=\bar{h}\) for which \(V(C) \neq V(\bar{C})\). It suffices to show a single case to refute the claim. To this end, let:

\[
\begin{aligned}
& R=5, \quad r=4, \quad h=6, \quad p=2 \\
& \bar{R}=4, \quad \bar{r}=2, \quad \bar{h}=6, \quad \bar{p}=1 \\
& V(C)=V(K)-V(Z)-2 V(A) \neq V(\bar{C})=V(\bar{K})-V(\bar{Z})-2 V(\bar{A}) \\
& \pi\left(\frac{4}{3} R^{3}-r^{2} h-2 R p^{2}+\frac{2}{3} p^{3}\right) \neq \pi\left(\frac{4}{3} \bar{R}^{3}-\bar{r}^{2} \bar{h}-2 \bar{R} \bar{p}^{2}+\frac{2}{3} \bar{p}^{3}\right) \\
& \pi\left(\frac{4}{3} \cdot 5^{3}-4^{2} \cdot 6-2 \cdot 5 \cdot 2^{2}+\frac{2}{3} \cdot 2^{3}\right) \neq \pi\left(\frac{4}{3} \cdot 4^{3}-2^{2} \cdot 6-2 \cdot 4 \cdot 1^{2}+\frac{2}{3} \cdot 1^{3}\right) \\
& 36 \pi \neq 54 \pi \\
& V(C) \neq V(\bar{C})
\end{aligned}
\]

Birgit is right.