Two airplanes leave an airport at the same time. The first flies 270km/h in a direction of 190 degrees. The second flies 150 km/h in a direction of 330 degrees. After 5 hours, how far apart are the planes?

To solve this, we can use vector analysis and the Law of Cosines to find the distance between the planes after 5 hours.

### Step-by-Step Solution:

1. **Calculate the Distance Traveled by Each Plane:**
   Each plane travels for 5 hours at its respective speed.
   
   - **First plane:**  
     \[
     \text{Distance} = 270 \, \text{km/h} \times 5 \, \text{hours} = 1350 \, \text{km}
     \]

   - **Second plane:**  
     \[
     \text{Distance} = 150 \, \text{km/h} \times 5 \, \text{hours} = 750 \, \text{km}
     \]

2. **Determine the Angle Between Their Directions:**
   - The direction of the first plane is 190 degrees.
   - The direction of the second plane is 330 degrees.

   To find the angle between them, calculate the absolute difference:
   \[
   \text{Angle between planes} = |330 - 190| = 140^\circ
   \]

3. **Apply the Law of Cosines:**
   Let \( a = 1350 \) km, \( b = 750 \) km, and the angle \( \theta = 140^\circ \). The distance \( d \) between the two planes after 5 hours is given by:
   \[
   d = \sqrt{a^2 + b^2 - 2ab \cdot \cos(\theta)}
   \]

   Substitute the values:
   \[
   d = \sqrt{1350^2 + 750^2 - 2 \cdot 1350 \cdot 750 \cdot \cos(140^\circ)}
   \]

4. **Calculate the Cosine Value and Solve:**
   Using \( \cos(140^\circ) \approx -0.7660 \):
   \[
   d = \sqrt{1350^2 + 750^2 - 2 \cdot 1350 \cdot 750 \cdot (-0.7660)}
   \]

   \[
   d = \sqrt{1822500 + 562500 + 2076750}
   \]

   \[
   d = \sqrt{4461750} \approx 2113.25 \, \text{km}
   \]

### Final Answer:
The distance between the two planes after 5 hours is approximately **2113.25 km**.

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Would you like a breakdown of each calculation step, or have any other questions?

Here are 5 related questions you might find helpful:
1. How would changing the direction of either plane affect the distance calculation?
2. What would the distance be if both planes traveled at the same speed but in different directions?
3. How could we calculate the bearing from one plane to the other after 5 hours?
4. What is the effect of time on the separation distance if both speeds are increased proportionally?
5. How can this problem be generalized for any two directions and speeds?

**Tip:** When solving vector problems, drawing a diagram can often help visualize the relative directions and angles involved.

Explore how to calculate the distance between two airplanes flying at different speeds and directions after 5 hours using vector analysis and the Law of Cosines. This solution involves trigonometry and vector principles, suitable for Grades 10-12 students.

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