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This is a classic problem involving the Law of Cosines to find the distance between two points based on the angles and side lengths in a non-right triangle.

### Given:
- \( AC = 62.5 \, \text{km} \)
- \( BC = 79.4 \, \text{km} \)
- Angle \( \angle ACB = 82^\circ \)

The goal is to find the distance \( AB \), which represents the distance between the two boats.

### Applying the Law of Cosines:
\[
AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(\angle ACB)
\]

Let's compute this step by step.The distance between the two boats, \( AB \), is approximately 94 km.

Therefore, the correct answer is:

**B. 94 km**

Would you like any further explanation or have additional questions?

Here are some related questions for practice:
1. What is the Law of Cosines, and how is it derived?
2. How would this problem change if the angle \( \angle ACB \) were 90 degrees?
3. Can the Law of Cosines be applied in right-angle triangles?
4. How can you verify your result using a different method?
5. How does changing the angle between the boats affect the distance between them?

**Tip:** Always remember to convert angles to radians when working with trigonometric functions in many calculators and programming environments.

Dos botes zarpan desde un puerto C en una dirección que forma un ángulo de 82° entre ellos. Cuando el bote A ha navegado 62.5 km, el bote B ha navegado 79.4 km. ¿Cuál es la distancia entre ellos?

Learn how to calculate the distance between two boats that traveled 62.5 km and 79.4 km respectively, with an angle of 82° between their paths. Using the Law of Cosines, this trigonometry problem is ideal for students in Grades 10-12.

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