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The problem describes the motion of a ship in two segments and asks for the final bearing of the ship from the pier. Let's break it down step by step:

1. **First Leg:**
   - The ship travels on a bearing of \( S 62^\circ E \) for 75 Nautical Miles (NM).
   - This means the ship moves southeast, forming an angle of \( 62^\circ \) east of due south.
   
2. **Second Leg:**
   - After turning around, the ship moves on a bearing of \( N 28^\circ E \) for 53 Nautical Miles.
   - This leg moves the ship northeast, forming an angle of \( 28^\circ \) east of due north.

### Steps to solve:
- **Convert bearings to Cartesian coordinates:** Bearings can be transformed into directions on an XY-plane where North corresponds to the positive y-axis, and East corresponds to the positive x-axis.
  
  - First leg:
    \[
    x_1 = 75 \sin(62^\circ), \quad y_1 = -75 \cos(62^\circ)
    \]
  - Second leg:
    \[
    x_2 = 53 \sin(28^\circ), \quad y_2 = 53 \cos(28^\circ)
    \]
  
- **Find the total displacement:** Add the x and y components from both legs to get the overall displacement vector.
  \[
  x_{\text{total}} = x_1 + x_2
  \]
  \[
  y_{\text{total}} = y_1 + y_2
  \]
  
- **Determine the bearing:** Using trigonometry, the final bearing of the ship from the pier can be found by calculating the angle of the displacement vector from the north-south axis.
  \[
  \theta = \tan^{-1}\left(\frac{x_{\text{total}}}{y_{\text{total}}}\right)
  \]
  If the displacement is in the first quadrant (both x and y positive), the bearing will be \( N \theta^\circ E \).

Let me calculate the values.The final bearing of the ship from the pier is approximately \( N 82.75^\circ E \). 

This means the ship is almost directly east of the pier with a slight deviation to the north.

Would you like further details or explanations on any part of this solution? Here are five related questions you might find interesting:

1. How do you calculate the bearing when given two or more legs of a journey?
2. Why is the tangent inverse function used to calculate the final bearing?
3. How would this calculation change if the journey took place on a curved surface (like Earth)?
4. What is the importance of converting bearings to Cartesian coordinates?
5. How would the solution change if the ship traveled westward instead?

**Tip:** Bearings are typically measured clockwise from the north. When calculating bearings from coordinates, always adjust the angle to fit this convention.

A ship leaves a pier on a bearing of S 62° E and travels for 75 NM. It then turns around and continues on a bearing N 28° E for 53 NM. Determine the bearing of the ship from the pier?

Determine the final bearing of a ship that travels 75 NM on a bearing of S 62° E, turns around, and continues 53 NM on a bearing N 28° E. This problem involves trigonometry and vector addition, making it ideal for students in Grades 9-12.

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