The problem provided can be solved using the Law of Cosines. Let’s break it down:

• A sailor sails from port A to port B, which is 100 km away.
• After arriving at port B, the sailor turns by an angle of 120 degrees and continues to port C, traveling 80 km.

The task is to find the distance between port A and port C.

Using the Law of Cosines:

The Law of Cosines formula is given by: $AC^2 = AB^2 + BC^2 - 2(AB)(BC) \cdot \cos(\theta)$ Where:

• $AB = 100 \, \text{km}$
• $BC = 80 \, \text{km}$
• $\theta = 120^\circ$

Let’s plug in the values and calculate the distance $AC$.

I'll calculate it now.The distance between port A and port C is approximately 156.2 km.

This value is closest to option (d) 160 km.

Would you like further details on the steps involved or clarification on the solution?

Here are five related questions for practice:

1. How would the calculation change if the angle were 90 degrees instead of 120?
2. What is the formula for the Law of Cosines and when is it applicable?
3. How would you approach the problem if the distances from port A to B and B to C were unknown?
4. Can you solve this problem graphically using vector addition?
5. How does the Law of Sines differ from the Law of Cosines?

Tip: The Law of Cosines is particularly useful for non-right triangles, where standard trigonometry (like the Pythagorean theorem) doesn't apply.