\section*{Problem \(5-330915=331015\)}

For a straight circular cylinder-shaped open-top tin can with base radius \(r\) and height \(h\), let \(A\) and \(B\) be the endpoints of a diameter of the base, with \(A\) outside and \(B\) inside the can. The thickness of the metal is negligible.

}

An ant moves from \(A\) to \(B\)

a) only along generatrix lines and a diameter of the base,

b) along the shortest path among all paths from \(A\) to \(B\) that do not leave the outer and inner lateral surfaces.

Determine a value of the ratio \(h: r\) for which the two paths described in a) and b) are of equal length!
}

The path described in a) has the length \(2 h + 2 r\).

If one imagines the lateral surface cut along the generatrix passing through \(A\) and unrolled into the drawing plane, the upper edge of the lateral surface turns into a segment \(s\) of length \(2 \pi \cdot r\). If one further imagines the inside of the unrolled lateral surface as a second copy on the back side of the drawing plane, which can now rotate around the segment \(s\) as an axis into the front side of the drawing plane, a rectangle with side lengths \(2 h\) and \(2 \pi \cdot r\) is formed, in which \(A\) is a corner and \(B\) is the midpoint of the side that does not contain \(A\) and is \(2 \pi \cdot r\) long (see figure; the base of the can has also been brought into the drawing plane).

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1136.jpg?height=432&width=744&top_left_y=1049&top_left_x=656)

In these transformations, the path lengths on the lateral surface do not change. A shortest possible path from \(A\) to \(B\) mentioned in b) must therefore run straight and thus, according to the Pythagorean theorem, have the length

\[
\sqrt{4 h^{2} + \pi^{2} r^{2}}
\]

The two paths are therefore of equal length if

\[
2 h + 2 r = \sqrt{4 h^{2} + \pi^{2} r^{2}}
\]

This is the case if

\[
\begin{aligned}
4 h^{2} + 8 h r + 4 r^{2} & = 4 h^{2} + \pi^{2} r^{2} \\
8 h & = (\pi^{2} - 4) \cdot r
\end{aligned}
\]

Thus, a sought value is determined as:

\[
h : r = \frac{\pi^{2} - 4}{8} \quad (\approx 0.7337)
\]