To solve this, let's break down the movement of the ship into two legs and use trigonometry to determine the distance and bearing.

Step 1: Break Down the Ship’s Path

1. Leg 1: The ship starts at port and travels 7.5 miles at a bearing of 28°.
2. Leg 2: It then turns and travels 4.1 miles due east to its destination.

The goal is to:

1. Determine the straight-line distance from the destination back to the port.
2. Calculate the bearing from the destination back to the port.

Step 2: Find the Coordinates of the Destination

Using trigonometry, let's determine the coordinates of the ship’s destination after each leg.

After Leg 1:

Since the ship travels 7.5 miles at a bearing of 28°, we can decompose this into:

• Horizontal (x) component: $x_1 = 7.5 \cdot \cos(28°)$
• Vertical (y) component: $y_1 = 7.5 \cdot \sin(28°)$

After Leg 2:

From this point, the ship travels due east by 4.1 miles:

• New x-coordinate: $x = x_1 + 4.1$
• New y-coordinate: $y = y_1$

Step 3: Calculate the Distance from Port to the Destination

Using the Pythagorean theorem for the straight-line distance $d$ from port to the destination: $d = \sqrt{x^2 + y^2}$

Step 4: Calculate the Bearing from Destination to Port

The bearing from the destination to the port can be found by calculating the angle back to the port using: $\theta = \arctan\left(\frac{y}{x}\right)$ Adjust this angle to express it as a standard bearing from north.

Let’s plug in the numbers.The calculations provide the following results:

1. Distance from the port to the ship's destination: 11.29 miles
2. Bearing from the destination back to the port: 198.18°

This means the ship is approximately 11.29 miles from the port, and if it were to return directly, it would need to travel at a bearing of 198.18°.

Would you like a detailed breakdown of any part of this solution, or do you have additional questions?

---

Here are 5 related questions:

1. How do you calculate bearings when the angles involve multiple quadrant shifts?
2. What would the path look like if the ship had traveled due west instead?
3. How can we determine the ship's path using vector components instead of trigonometry?
4. What if the ship traveled north after the initial leg instead of east—how would the calculations change?
5. How does trigonometric rounding affect results in real-world navigation?

Tip: Always ensure bearings are adjusted to standard compass bearings, as navigation relies on consistent directional notation.