Task 4. (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 4 km/h along a road each arbitrarily chooses. Determine the probability that the distance between them after one hour will be more than 6 km.

#
# Solution.

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

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-28.jpg?height=409&width=292&top_left_y=181&top_left_x=882)

Fig. 1

By the cosine theorem: \( x^{2}=4^{2}+4^{2}-2 \cdot 4 \cdot 4 \cos 45^{\circ}=16(2-\sqrt{2}), x=4 \sqrt{2-\sqrt{2}} \), which is less than 6.

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

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-28.jpg?height=351&width=385&top_left_y=861&top_left_x=844)

4

Fig. 2

By the Pythagorean theorem: \( x^{2}=4^{2}+4^{2}, x=4 \sqrt{2} \), which is less than 6.

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

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-28.jpg?height=221&width=417&top_left_y=1477&top_left_x=888)

Fig. 3

By the cosine theorem: \( x^{2}=4^{2}+4^{2}-2 \cdot 4 \cdot 4 \cos 135^{\circ}=16(2+\sqrt{2}), x=4 \sqrt{2+\sqrt{2}} \), which is greater than 6.

Thus, the distance will be greater than 6 km if the geologists choose different roads, specifically those two 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 apart), i.e., \( n=8 \cdot 3=24 \).

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

Answer. \(\mathbf{0.375.}\)