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

For an open-top cylindrical can made of sheet metal with a base radius \(r\) and height \(h\), let \(A\) and \(B\) be the endpoints of a diameter of the base. Here, \(A\) lies outside and \(B\) inside the can. The thickness of the sheet 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 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-1625.jpg?height=425&width=736&top_left_y=1669&top_left_x=660)

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:

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