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To solve this problem, we need to find the distance between two planes after they have flown in different directions for a certain amount of time.

### Given Data:
1. **Plane 1**:
   - Speed: 270 km/h
   - Direction: 330°
   - Time: 5 hours

2. **Plane 2**:
   - Speed: 150 km/h
   - Direction: 190°
   - Time: 5 hours

### Step-by-Step Solution:
1. **Calculate the Distances Traveled by Each Plane**:
   - Distance \( d_1 \) traveled by Plane 1:  
     \( d_1 = \text{Speed} \times \text{Time} = 270 \, \text{km/h} \times 5 \, \text{h} = 1350 \, \text{km} \)
   - Distance \( d_2 \) traveled by Plane 2:  
     \( d_2 = \text{Speed} \times \text{Time} = 150 \, \text{km/h} \times 5 \, \text{h} = 750 \, \text{km} \)

2. **Determine the Angle Between Their Paths**:
   - Plane 1 heads in the direction of 330°, and Plane 2 heads in the direction of 190°.
   - The difference between these directions is:  
     \( \theta = |330° - 190°| = 140° \)

3. **Use the Law of Cosines**:
   To find the distance \( D \) between the two planes after 5 hours, use the formula:
   \[
   D = \sqrt{d_1^2 + d_2^2 - 2 \cdot d_1 \cdot d_2 \cdot \cos(\theta)}
   \]
   Substituting the values:
   \[
   D = \sqrt{1350^2 + 750^2 - 2 \cdot 1350 \cdot 750 \cdot \cos(140°)}
   \]

4. **Calculate \( \cos(140°) \) and Plug into the Equation**:
   - Using \( \cos(140°) \approx -0.7660 \):
   \[
   D = \sqrt{1350^2 + 750^2 + 2 \cdot 1350 \cdot 750 \cdot 0.7660}
   \]
   \[
   D = \sqrt{1822500 + 562500 + 2071350}
   \]
   \[
   D = \sqrt{4456350} \approx 2111 \, \text{km}
   \]

### Final Answer:
After 5 hours, the planes are approximately **2111 km** apart.

Would you like further details, or do you have any questions?

### Related Questions:
1. How would the distance change if both planes flew in the same direction?
2. What would happen if the time was reduced to 3 hours instead of 5?
3. How do we interpret angles greater than 180° in navigational problems?
4. How does the Law of Cosines work in calculating distances in vector problems?
5. What other methods can be used to calculate distances if directions are given in 3D space?

### Tip:
Always break down navigation problems into vector components to make calculations easier.

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

Learn how to calculate the distance between two planes traveling in different directions using trigonometry and the Law of Cosines. Given speeds, directions, and time, find the separation distance after 5 hours. This problem is suitable for Grade 11-12 students and covers key concepts in trigonometry and vectors.

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