A ship is sailing due north. At a certain​ point, the bearing of a lighthouse 4.94.9 km away is N38.738.7degrees°E. Later​ on, the captain notices that the bearing of the lighthouse has become  Upper S 34.6 degrees E.S34.6°E. How far did the ship travel between the two observations of the​ lighthouse?
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Part 1
The ship travelled  4.74.7 km between the two observations.
​(Do not round until the final answer. Then round to the nearest tenth as​ needed.)

To solve this problem, we need to use trigonometry, particularly the Law of Sines or Law of Cosines, to find the distance the ship has traveled between the two observations of the lighthouse. Here's a step-by-step solution:

1. **Understanding the Bearings:**
   - The initial bearing of the lighthouse from the ship is N 38.7° E, meaning the lighthouse is 38.7° east of due north.
   - The final bearing of the lighthouse from the ship is S 34.6° E, meaning the lighthouse is 34.6° east of due south.

2. **Sketching the Situation:**
   - Draw a north-south line representing the ship's path.
   - At the starting point, draw a line representing the direction to the lighthouse at N 38.7° E.
   - At the final point, draw a line representing the direction to the lighthouse at S 34.6° E.
   - The distance from the starting point to the lighthouse is 4.9 km.
   - The distance from the final point to the lighthouse is also 4.9 km.

3. **Using Trigonometry:**
   - The angle between the two bearings can be calculated by adding the angles measured from north and south respectively because they are on opposite sides of the north-south line:
     \[
     \theta = 38.7^\circ + 34.6^\circ = 73.3^\circ
     \]

4. **Applying the Law of Cosines:**
   - Let \( d \) be the distance traveled by the ship between the two observations.
   - Using the Law of Cosines:
     \[
     d^2 = 4.9^2 + 4.9^2 - 2 \cdot 4.9 \cdot 4.9 \cdot \cos(73.3^\circ)
     \]
   - Calculate the value:
     \[
     d^2 = 4.9^2 + 4.9^2 - 2 \cdot 4.9 \cdot 4.9 \cdot \cos(73.3^\circ)
     \]
     \[
     d^2 = 24.01 + 24.01 - 2 \cdot 24.01 \cdot \cos(73.3^\circ)
     \]
     \[
     d^2 = 48.02 - 48.02 \cdot \cos(73.3^\circ)
     \]
     \[
     d^2 = 48.02 - 48.02 \cdot 0.289
     \]
     \[
     d^2 = 48.02 - 13.8738
     \]
     \[
     d^2 = 34.1462
     \]
     \[
     d = \sqrt{34.1462} \approx 5.84 \, \text{km}
     \]

Thus, the ship traveled approximately 5.8 km between the two observations of the lighthouse.

Would you like more details on any part of this solution, or do you have any other questions?

Here are some questions you might want to ask:
1. How do bearings work in navigation?
2. What is the Law of Sines and how is it used in navigation problems?
3. Can you explain the Law of Cosines in more detail?
4. How do you convert between degrees and radians?
5. What are some common mistakes to avoid when solving trigonometry problems?

**Tip:** Always sketch the problem when dealing with bearings and navigation; visualizing the situation helps in understanding the relationships between the angles and distances.

Learn how to calculate the distance traveled by a ship using trigonometry, specifically the Law of Cosines, and interpreting bearings. This problem involves determining the ship's path based on initial and final bearings of a lighthouse. Suitable for high school students and those interested in navigation problems.

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