2. (13 points) In a hall, the floor has dimensions $4 \times 5 \mathrm{~m}^{2}$, and the ceiling height is 4 m. On the ceiling, in one corner, sits a fly named Masha, and in the opposite corner of the ceiling, a spider named Petya. Masha set off on foot to visit Petya by the shortest route, but with a stop on the floor. Find the length of the path she traveled.
Answer: $\sqrt{145}$ m.

Solution. Let the fly sit at vertex $M$, and the spider sit at vertex $\Pi$ (Fig. 1.).

![](https://cdn.mathpix.com/cropped/2024_05_06_b6ca313a2688e0b12313g-01.jpg?height=526&width=554&top_left_y=1242&top_left_x=845)

There are 4 choices of paths for Masha.

Masha can start her journey from the face (wall) $B M C_{1} C$, walk across the floor, and finish her journey on the wall $D A A_{1} \Pi$. In this case, her shortest path to Petya is visible on the net (Fig. 2.).

![](https://cdn.mathpix.com/cropped/2024_05_06_b6ca313a2688e0b12313g-01.jpg?height=617&width=354&top_left_y=2030&top_left_x=957)

Fig. 2.

In this case, the length of the path is $\sqrt{12^{2}+5^{2}}=\sqrt{169}=13$ m (Pythagorean theorem).

In the second case, Masha can start her journey from the face $B M A_{1} A$, and then her shortest path to Petya is visible on the net (Fig. 3.).

![](https://cdn.mathpix.com/cropped/2024_05_06_b6ca313a2688e0b12313g-02.jpg?height=497&width=668&top_left_y=391&top_left_x=800)

Fig. 3.

In this case, the length of the path is $\sqrt{9^{2}+8^{2}}=\sqrt{145}$ m.

In the third case, Masha will start, as in the first case, from the wall $B M C_{1} C$, but will finish her movement on the wall $D \Pi C_{1} C$. In this case, the length of the path is, as in the second case, $\sqrt{9^{2}+8^{2}}=\sqrt{145}$ m.

In the fourth case, Masha will start, as in the second case, from the wall $B M A_{1} A$, and finish on the face $D \Pi C_{1} C$. The length of the path in this case is $\sqrt{4^{2}+13^{2}}=\sqrt{185}$ m.

The shortest path for Masha is $\sqrt{145}$ m.

Grading criteria. Correct complete solution - 13 points. A non-shortest path (one of those provided in the solution) is proposed, but its length is correctly found - 6 points. If a path that is not a straight line segment on any net is proposed and declared the shortest, then 0 points for the problem. The correct path is chosen, but not justified as the shortest, with the correct answer - 11 points. There are arithmetic errors in a correct solution approach - minus 2 points.