# Task 1. 15 points

The distances between the piers on the island, where the loading and unloading of fish catches take place, are 4 km, $\sqrt{13}$ km, and 3 km, respectively. A refrigeration unit, located within the transport zone (a triangle with vertices at the loading-unloading points), is connected to the piers by straight roads. What is the minimum distance that a refrigerated truck will travel if it needs to make three trips to each pier in sequence, each time leaving from the refrigeration unit and returning back.

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# Solution

The solution to the problem reduces to finding the minimum distance from some point inside a triangle with sides 4 km, $\sqrt{13}$ km, and 3 km, and then multiplying this distance by 2 (the refrigerated truck travels to and from the refrigerator to each pier for loading and unloading without repeating ports).

Let in triangle $\mathrm{ABC}$, the base $\mathrm{AB}=3, \mathrm{AC}=4, \mathrm{BC}=\sqrt{13}$. From the cosine theorem, we get that $\angle A=60^{\circ}$. E is a point inside the triangle such that EA+EB+EC is minimized. $\triangle \mathrm{AEC}$ is rotated 60 degrees around A outside the original triangle so that C lands at $\mathrm{C}_{1}$, and $\mathrm{E}$ lands at $\mathrm{E}_{1}$. Note that $\mathrm{CC}_{1}$ is parallel to $\mathrm{AB}$, and $\triangle \mathrm{AEE}_{1}$ is an equilateral triangle. Clearly, $\mathrm{EA}+\mathrm{EB}+\mathrm{EC}=\mathrm{C}_{1} \mathrm{E}_{1}+\mathrm{E}_{1} \mathrm{E}+\mathrm{EB}$. The minimum value of the latter sum is the length of segment $\mathrm{C}_{1} \mathrm{~B}$. Thus, the original problem is reduced to finding the length of $\mathrm{C}_{1} \mathrm{~B}$, the side of $\triangle \mathrm{ABC}_{1}$, where $\mathrm{AC}_{1}=\mathrm{AC}=4, \mathrm{AB}=3, \angle C_{1} A B=120^{\circ}$. By the cosine theorem, $C_{1} B=\sqrt{A C^{2}+A B^{2}-2 A C \cdot A B \cdot \cos 120^{\circ}}=\sqrt{4^{2}+3^{2}-2 \cdot 4 \cdot 3 \cdot(-0.5)}=\sqrt{37}$. Since the refrigerated truck travels to and from, the obtained value needs to be multiplied by 2.

Answer: $2 \sqrt{37}$

## Criteria:

- The correct solution to the problem and the correct answer are provided - 15 points,
- The correct solution to the problem, but the wrong answer is provided - 14 points,
- An error in the solution, but the correct answer is provided - 8 points,
- The solution is incorrect or absent, but the correct answer is provided - 1 point,
- The solution is incorrect or absent, the wrong answer is provided, or no answer is given - 0 points.