A ship travels on a bearing S60°W so that it is 90 km west of its starting position. How far had it travelled on this bearing? How far is it to the south of its starting position?

To solve the problem, we'll break it into parts. The ship travels on a bearing of \( S60^\circ W \), which means it is heading 60 degrees west of due south.

### Part 1: Understanding the Direction and Components

- **Bearing \( S60^\circ W \)**: This indicates that the ship is moving in a direction 60 degrees west of south.
- Given that the ship is 90 km west of its starting position, we need to find:
  1. The total distance traveled by the ship.
  2. The distance the ship is to the south of its starting position.

### Part 2: Setting Up the Right Triangle

Given the bearing and the problem statement, we can create a right triangle where:
- The westward distance (horizontal leg) is 90 km.
- The distance traveled by the ship forms the hypotenuse of the triangle.
- The southward distance (vertical leg) is what we need to calculate.

Let's define:
- \( D \) as the total distance traveled by the ship (hypotenuse).
- \( S \) as the distance southward (vertical leg).
- \( W = 90 \) km as the distance westward (horizontal leg).

### Part 3: Calculate the Total Distance Traveled

Using trigonometry:
- The westward component is related to the total distance by the cosine of the angle:
\[
\cos(60^\circ) = \frac{\text{Westward Distance}}{\text{Total Distance}}
\]
\[
\cos(60^\circ) = \frac{90}{D}
\]
Since \(\cos(60^\circ) = 0.5\):
\[
0.5 = \frac{90}{D}
\]
\[
D = \frac{90}{0.5} = 180 \text{ km}
\]
Thus, the ship traveled **180 km** on this bearing.

### Part 4: Calculate the Southward Distance

Using trigonometry:
- The southward component is related to the total distance by the sine of the angle:
\[
\sin(60^\circ) = \frac{S}{D}
\]
\[
S = D \times \sin(60^\circ)
\]
\[
S = 180 \times \sin(60^\circ)
\]
Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866\):
\[
S = 180 \times 0.866 \approx 155.88 \text{ km}
\]
Thus, the ship is approximately **155.88 km** south of its starting position.

### Final Answers
- The ship traveled a total distance of **180 km** on this bearing.
- The ship is **155.88 km** south of its starting position.

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

### Follow-up Questions
1. How would the calculations change if the ship traveled on a bearing \( S30^\circ E \) instead?
2. What would be the ship's total distance if it ended up 120 km south instead of 90 km west?
3. How can you verify the calculated distances using the Pythagorean theorem?
4. What would be the result if the bearing was \( S45^\circ W \) instead of \( S60^\circ W \)?
5. How would you calculate the eastward component if the ship was traveling on a bearing \( N45^\circ E \)?

### Tip
When dealing with bearings and directions, always break down the movement into its horizontal and vertical components. This simplifies the calculation process.

Learn how to solve a trigonometry problem involving a ship traveling on a bearing S60°W. Understand the ship's total distance traveled and its southward distance from the starting position using trigonometric functions. Suitable for students in Grades 9-12.

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