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This question is about bearings and the Law of Sines to solve for the bearing and distance of a ship from a lighthouse.

### Question Summary:
We are given:
- A lighthouse \( L \) and two ships \( K \) and \( M \).
- The bearing of \( L \) from \( K \) is 155°.
- The distance between \( L \) and \( K \), \( KL \), is 112 km.
- The bearing of \( K \) from \( M \) is 010°.
- Angle \( KML = 96^\circ \).

The task is to find:
1. The bearing of \( M \) from \( L \).
2. The distance of \( M \) from the lighthouse \( L \).

### Step-by-Step Explanation:

#### 1. Find the Bearing of M from L:
We are asked to calculate the bearing of \( M \) from \( L \).

- From the diagram, we can infer some key angles. The bearing of \( K \) from the lighthouse \( L \) is 155°. This forms part of a triangle where we are trying to find the bearing of \( M \) from \( L \).
- The angle at \( M \) is given as \( 96^\circ \), and from the diagram, angle \( LKM \) = \( 35^\circ \).
  
Using the sum of interior angles of a triangle (180°):
\[
\text{Angle } KLM = 180^\circ - 96^\circ - 35^\circ = 49^\circ.
\]

The bearing of \( M \) from \( L \) is then calculated by subtracting the known angles from 360°:
\[
\text{Bearing of } M \text{ from } L = 360^\circ - (49^\circ + 25^\circ) = 286^\circ.
\]

#### 2. Find the Distance of M from the Lighthouse L:
To find the distance of \( M \) from the lighthouse \( L \), we apply the **Law of Sines**.

The Law of Sines formula is:
\[
\frac{ML}{\sin(\theta_{ML})} = \frac{KL}{\sin(\theta_{KL})}
\]
where:
- \( ML \) is the distance of \( M \) from \( L \) (which we are solving for),
- \( KL = 112 \, \text{km} \),
- \( \theta_{ML} = 35^\circ \) (angle opposite \( ML \)),
- \( \theta_{KL} = 96^\circ \) (angle opposite \( KL \)).

Rearranging for \( ML \):
\[
ML = \frac{112 \times \sin(35^\circ)}{\sin(96^\circ)}.
\]

Now, we calculate the value:
\[
ML = \frac{112 \times \sin(35^\circ)}{\sin(96^\circ)} \approx 64.6 \, \text{km}.
\]

Thus, the distance of ship \( M \) from the lighthouse \( L \) is approximately **64.6 km**.

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### To summarize:
1. The bearing of \( M \) from \( L \) is **286°**.
2. The distance of \( M \) from \( L \) is **64.6 km**.

Would you like further details or have any other questions?

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### Five related questions:
1. How do bearings relate to angles in geometry?
2. What is the Law of Sines, and when is it used?
3. Can the Law of Sines be applied to non-right triangles?
4. How would we solve this problem if \( KLM \) were a right triangle?
5. What role do bearings play in navigation and geometry?

### Tip:
When solving for bearings, always ensure you are working with a full 360° circle, and bearings are measured clockwise from the North.

The diagram shows the positions of a lighthouse L, and two ships, K and M. The bearing of L from K is 155°, and KL = 112 km. The bearing of K from M is 010° and angle KML = 96°. Find the bearing and distance of ship M from the lighthouse L.

Solve a navigation problem involving bearings and the Law of Sines to find the bearing of ship M from lighthouse L and the distance ML. The question involves bearings of K, L, and M, as well as the application of the Law of Sines. Suitable for students in Grades 10-12.

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