a ship leaves port with a bearing of S 42deg W after traveling 7 miles, the ship turns 90deg and travels on a bearing of N 52deg W for 10 miles at that time, what is the bearing of the ship to the port?

just convert each distance/direction to x- and y-components, and add them up. Any problems?

To determine the bearing of the ship to the port, we need to visualize the movement of the ship and apply some trigonometry.

1. Start by drawing a diagram. Draw a line segment to represent the ship's initial movement of 7 miles on a bearing of S 42° W. Mark the starting point as the port. Label this line as AB.

2. From point B, draw a line segment perpendicular to AB. This line segment represents the ship's 90° turn. Label this line as BC.

3. Extend line segment BC 10 miles in the direction of N 52° W. Label the endpoint of this line as C, and connect C back to A. This new line segment represents the ship's final position.

4. To find the bearing of the ship to the port, we need to find the angle that segment CA forms with the north direction (points towards the top of the diagram). We'll call this angle α.

5. Using trigonometry, we can find angle α. First, focus on the triangle formed by segment AC, the north direction line (top of the diagram), and the vertical line connecting C to the north direction line.

6. In this triangle, we have the opposite side (10 miles) and the adjacent side (7 miles). To find angle α, we need to use the tangent function: tan(α) = opposite/adjacent = 10/7.

7. Use a scientific calculator or the inverse tangent function (tan⁻¹) to find α. α ≈ tan⁻¹(10/7) ≈ 57.6°.

8. Now, to determine the bearing of the ship to the port, we need to subtract α from 180° (due to the clockwise direction). Bearing of the ship to the port = 180° - 57.6° = 122.4°.

Therefore, the bearing of the ship to the port is approximately N 122.4° E.