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Let's solve the parts (b), (c), and (d) of the given problem based on the provided information.

### Given:
- Harrison walked \( XY = 450 \, \text{m} \) from \( X \) to \( Y \) on a bearing of \( 214^\circ \).
- He then walked \( YZ = 520 \, \text{m} \) to \( Z \), which is \( 610 \, \text{m} \) due west of \( X \).

### Part (b): Use the Cosine Rule to find \( \angle ZXY \) (denoted as \( \theta \)), correct to the nearest degree.

To solve for \( \angle ZXY \), we'll use the Cosine Rule:
\[
\cos(\theta) = \frac{XZ^2 + XY^2 - YZ^2}{2 \times XZ \times XY}
\]
Where:
- \( XZ = 610 \, \text{m} \)
- \( XY = 450 \, \text{m} \)
- \( YZ = 520 \, \text{m} \)

Let's substitute these values into the formula:

\[
\cos(\theta) = \frac{610^2 + 450^2 - 520^2}{2 \times 610 \times 450}
\]

Now I'll calculate the value of \( \theta \).

### Part (c): Find the bearing of \( Y \) from \( Z \).

The bearing from \( Z \) to \( Y \) can be found by considering the angle \( \angle ZXY \) that we just calculated. The bearing of \( Y \) from \( Z \) is:

\[
\text{Bearing} = \theta + 180^\circ
\]

### Part (d): Find the bearing of \( Z \) from \( Y \).

The bearing from \( Y \) to \( Z \) can be found by subtracting \( 180^\circ \) from the bearing of \( Y \) from \( Z \), as it would be directly opposite.

Let me perform these calculations.### Solution:

**Part (b):**  
The angle \( \angle ZXY \) (denoted as \( \theta \)) is approximately \( 56^\circ \) (rounded to the nearest degree).

**Part (c):**  
The bearing of \( Y \) from \( Z \) is approximately \( 236^\circ \) (rounded to the nearest degree).

**Part (d):**  
The bearing of \( Z \) from \( Y \) is approximately \( 56^\circ \) (rounded to the nearest degree).

Do you want more details or have any questions?

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Here are some related questions you might find interesting:
1. How would the values change if the distance \( XZ \) was different?
2. Can you find the distance \( XZ \) using the Sine Rule?
3. How would the solution change if Harrison walked \( 520 \, \text{m} \) to the north instead of west?
4. What is the significance of using the Cosine Rule in this type of problem?
5. How would the angle \( \theta \) change if the bearing from \( X \) to \( Y \) was \( 120^\circ \)?

**Tip:** When solving bearing problems, always double-check the direction and angle measurements to ensure they correspond to the correct quadrant.

Learn how to solve a trigonometry problem involving bearings and the Cosine Rule. Calculate angles and bearings between points X, Y, and Z. Detailed step-by-step solution provided.

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