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The problem you provided involves the concept of bearings and vectors. Here's a breakdown of the key elements:

- A radar station detects two planes:
  - One plane is 18.5 km away on a bearing of 24°.
  - The second plane is 10.0 km away on a bearing of 180°.
  
You are asked to:
1. Find the displacement between the two planes.
2. Determine the bearing of the second plane from the first plane.

Let’s begin solving:

### 1. Finding the Displacement Between the Two Planes

This can be solved using the Law of Cosines, given that you are working with a non-right triangle where:
- One side (from the radar to plane 1) is 18.5 km.
- Another side (from the radar to plane 2) is 10.0 km.
- The angle between these sides is the difference in their bearings, which is \(180° - 24° = 156°\).

Using the Law of Cosines:

\[
d^2 = 18.5^2 + 10.0^2 - 2 \cdot 18.5 \cdot 10.0 \cdot \cos(156^\circ)
\]

Now, let me compute this for you.The displacement between the two planes is approximately **27.9 km**.

### 2. Bearing of the Second Plane from the First

To calculate the bearing of the second plane from the first plane, we need to use the Law of Sines or another vector-based method to determine the angle. Here's how:

- Use the Law of Sines to find the angle at the radar between the displacement and the line to the second plane.
- Then, add or subtract from the initial bearing to find the bearing of the second plane from the first plane.

I will calculate that next.It seems like I can't do more advanced data analysis right now. Please try again later.

However, I can guide you through calculating the bearing manually if you'd like to do it:

1. Use the Law of Sines:  
   \[
   \sin(B) = \frac{10.0 \cdot \sin(156^\circ)}{27.9}
   \]
   Solve for angle \(B\), which is the angle at the second plane.

2. Add this angle to the original bearing of 24° to get the final bearing of the second plane from the first.

If you have any questions or would like more help, feel free to ask!

Here are 5 related questions to explore:
1. How does the Law of Cosines help in solving vector problems?
2. What other methods can be used to solve for the distance between two points in navigation?
3. How is the Law of Sines applied in non-right-angled triangles?
4. Can you explain how bearings are used in navigation and aviation?
5. How does the bearing angle relate to true north and relative positioning?

**Tip:** When solving problems involving bearings and vectors, always draw the situation to visualize the angles clearly!

A radar station notes the bearing of two planes, one 18.5 km away on a bearing of 24°, and the other 10.0 km away on a bearing of 180°. Find the displacement of the second plane from the first.

Calculate the displacement between two planes detected by a radar station, with distances of 18.5 km and 10.0 km, and bearings of 24° and 180°. Learn how to apply the Law of Cosines and Law of Sines for accurate results.

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