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.