Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists set out on a journey from their cabin at a speed of 5 km/h along a road each arbitrarily chooses. Determine the probability that the distance between them after one hour will be more than 8 km.
# Solution.

Let's find the distance between the geologists after 1 hour if they are walking on adjacent roads (Fig. 2).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-13.jpg?height=406&width=292&top_left_y=985&top_left_x=882)

Fig. 2

By the cosine theorem: \( x^{2}=5^{2}+5^{2}-2 \cdot 5 \cdot 5 \cos 45^{\circ}=25(2-\sqrt{2}), x=5 \sqrt{2-\sqrt{2}} \), which is less than 8.

Let's find the distance between the geologists after 1 hour if they are walking on roads that are one road apart (Fig. 3).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-13.jpg?height=400&width=394&top_left_y=1662&top_left_x=842)

Fig. 3

By the Pythagorean theorem: \( x^{2}=5^{2}+5^{2}, x=5 \sqrt{2} \), which is less than 8.

Let's find the distance between the geologists after 1 hour if they are walking on roads that are two roads apart (Fig. 4).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-13.jpg?height=219&width=417&top_left_y=2272&top_left_x=885)

Fig. 4

By the cosine theorem: \( x^{2}=5^{2}+5^{2}-2 \cdot 5 \cdot 5 \cos 135^{\circ}=25(2+\sqrt{2}), x=5 \sqrt{2+\sqrt{2}} \), which is greater than 8.

Thus, the distance will be greater than 8 km if the geologists choose different roads, specifically those two roads apart.

Possible outcomes. The first geologist can choose any of the 8 roads, and the second geologist can also choose any of the 8 roads, i.e., \( n=8 \cdot 8=64 \).

Favorable outcomes. The first geologist can choose any of the 8 roads, and the second geologist can choose only 3 (not the same road, not adjacent roads, and not one road apart), i.e., \( n=8 \cdot 3=24 \).

Then, \( P=\frac{24}{64}=0.375 \).

Answer. \( 0.375 \).