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 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. 1).

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-23.jpg?height=405&width=292&top_left_y=183&top_left_x=882)

Fig. 1

By the Law of Cosines: \( x^{2}=5^{2}+5^{2}-2 \cdot 5 \cdot 5 \cos 45^{\circ}=25(2-\sqrt{2}) \),

\[
x=5 \sqrt{2-\sqrt{2}}, \text { which is less than } 8 .
\]

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

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-23.jpg?height=388&width=374&top_left_y=888&top_left_x=858)

Fig. 2

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 separated by two roads (Fig. 3).

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-23.jpg?height=235&width=419&top_left_y=1499&top_left_x=884)

Fig. 3

By the Law of Cosines: \( x^{2}=5^{2}+5^{2}-2 \cdot 5 \cdot 5 \cos 135^{\circ}=25(2+\sqrt{2}) \),

\[
x=5 \sqrt{2+\sqrt{2}}, \text { which is greater than } 8 .
\]

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

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 roads separated by one road), i.e., \( n=8 \cdot 3=24 \).

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

Answer. \( 0.375 \).