Task 4. (20 points) In the center of a circular field stands a geologists' cabin. From it, 6 straight roads extend, dividing the field into 6 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.

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# 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_e9b4fbbbcf148e16432dg-33.jpg?height=329&width=343&top_left_y=1466&top_left_x=868)

$60^{\circ}$

Fig. 2

Since the triangle is equilateral, $x=5$, 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 apart (Fig. 3).

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-33.jpg?height=417&width=460&top_left_y=2093&top_left_x=815)

Fig. 3

$\frac{y}{2}=5 \sin 60^{\circ}=\frac{5 \sqrt{3}}{2}, y=5 \sqrt{3}$, which is greater than 8.

Thus, the distance will be greater than 8 km if the geologists do not choose the same or adjacent roads.

Possible outcomes. The first geologist can choose any of the 6 roads, and the second geologist can also choose any of the 6 roads, i.e., $n=6 \cdot 6=36$.

Favorable outcomes. The first geologist can choose any of the 6 roads, and the second geologist can choose only 3 (not the same or adjacent roads), i.e., $n=6 \cdot 3=18$.

Then, $P=\frac{18}{36}=0.5$.

Answer. 0.5.