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To solve this problem, we need to find the displacement between the two planes based on the given bearings and distances.

Here’s a breakdown:

### Step 1: Understanding the setup
- Plane 1 is at a distance of 18.5 km from the radar station at a bearing of 24.0°.
- Plane 2 is at a distance of 10.0 km from the radar station at a bearing of 180.0°.

This forms a triangle with the radar station as the common point between the two planes.

### Step 2: Using the Law of Cosines
The angle between the bearings of the two planes is \( \theta = 180^\circ - 24^\circ = 156^\circ \). We use this information to apply the Law of Cosines.

The Law of Cosines formula is:
\[
d^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\]
Where:
- \( d \) is the distance between the two planes (what we are trying to find).
- \( a = 18.5 \, \text{km} \) is the distance of Plane 1 from the radar.
- \( b = 10.0 \, \text{km} \) is the distance of Plane 2 from the radar.
- \( \theta = 156^\circ \) is the angle between the two planes.

### Step 3: Solving for \( d \)
Let me compute this.The distance between the two planes is approximately **27.9 km**.

Would you like any more details or have further questions?

Here are five related questions you could explore:
1. How does the Law of Cosines relate to the Pythagorean theorem for right triangles?
2. How would the calculation change if the bearing angles were different?
3. How can you determine the displacement using trigonometric functions like sine or tangent?
4. What are real-world applications of bearings in navigation aside from aviation?
5. How can you verify the accuracy of a navigation system's bearings?

**Tip:** Always ensure to use the correct units and angle mode (degrees or radians) when applying trigonometric functions in calculations.

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°. Find the displacement of the second plane from the first.

Learn how to find the displacement between two planes using bearings and the Law of Cosines. This problem involves applying trigonometric principles to determine the distance between two planes based on their bearings and distances from a common radar station. Suitable for high school students in Grades 10-12.

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