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The problem in the image involves navigation and bearing, with two planes at different distances from a radar station. The key details provided are:

1. A radar station detects two planes:
   - One is 18.5 km away on a bearing of 24.0°.
   - The other is 10.0 km away on a bearing of 180.0°.
2. The distance between the two planes is given as 27.9 km.

We need to find:
1. **The bearing of the second plane from the first plane** (angle between them from the perspective of the first plane).

### Step-by-Step Solution:

#### Step 1: Analyze the Geometry
This is a classic triangle problem where we are given two sides and one angle:
- Side \( a = 18.5 \, \text{km} \) (distance from the radar to the first plane)
- Side \( b = 10.0 \, \text{km} \) (distance from the radar to the second plane)
- Side \( c = 27.9 \, \text{km} \) (distance between the two planes)

The angle between the two planes from the radar station is \( 180^\circ - 24^\circ = 156^\circ \).

#### Step 2: Use the Law of Cosines
We can use the law of cosines to find the angle at the radar station (angle \( C \)):

\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab}
\]

Substituting the values:

\[
\cos C = \frac{18.5^2 + 10.0^2 - 27.9^2}{2 \times 18.5 \times 10.0}
\]

We will compute this value to find angle \( C \), and then use this to find the bearing of the second plane from the first.

Let me calculate that.The angle at the radar station between the two planes (angle \( C \)) is approximately \( 155.3^\circ \).

#### Step 3: Bearing of the Second Plane from the First
To find the bearing of the second plane from the first, we can add the angle between the two planes to the bearing of the first plane. Since the first plane is on a bearing of \( 24.0^\circ \), the bearing of the second plane from the first is:

\[
\text{Bearing from first plane} = 24.0^\circ + 155.3^\circ = 179.3^\circ
\]

Thus, the bearing of the second plane from the first is approximately **179°**.

Would you like to explore any more details or have any other questions?

Here are 5 related questions:
1. How is the Law of Cosines derived and why is it used in this problem?
2. What are the limitations of the Law of Cosines in navigation problems?
3. How would the problem change if one of the bearings was missing?
4. Can the Law of Sines be used instead of the Law of Cosines in this scenario? Why or why not?
5. How does understanding bearings help in practical navigation?

**Tip:** Always ensure bearings are measured clockwise from true north to avoid confusion with angles in geometry.

In navigation, one method of expressing bearing is to give a single angle measured clockwise from due north. A radar station notes the bearing of two planes, one 18.5 km away on a bearing of 24.0°, and the other 10.0 km away on a bearing of 180.0°. The distance between the planes is 27.9 km. Find the bearing of the second plane from the first.

Solve a navigation problem involving bearings and distances between two planes. Using the Law of Cosines, calculate the bearing of the second plane from the first, given that the planes are 18.5 km and 10.0 km away from a radar station with a distance of 27.9 km between them.

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