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To find how far apart the two planes are after 5 hours, we can approach this as a vector problem or apply the law of cosines, as the directions of their flights create an angle between them. Here’s how to solve it step-by-step:

### Given Information:
1. **Plane 1** flies at **270 km/h** in a direction of **330°**.
2. **Plane 2** flies at **150 km/h** in a direction of **190°**.
3. **Time flown** for both planes is **5 hours**.

### Step 1: Calculate Distances Traveled
After 5 hours:
- **Plane 1’s distance** = \( 270 \, \text{km/h} \times 5 \, \text{hours} = 1350 \, \text{km} \)
- **Plane 2’s distance** = \( 150 \, \text{km/h} \times 5 \, \text{hours} = 750 \, \text{km} \)

### Step 2: Determine the Angle Between Their Paths
The difference in direction angles is:
- \( 330^\circ - 190^\circ = 140^\circ \)

Thus, the angle between the paths of the planes is \( 140^\circ \).

### Step 3: Apply the Law of Cosines
Using the Law of Cosines to find the distance \( d \) between the two planes:
\[
d = \sqrt{1350^2 + 750^2 - 2 \times 1350 \times 750 \times \cos(140^\circ)}
\]

#### Calculation:
1. \( 1350^2 = 1822500 \)
2. \( 750^2 = 562500 \)
3. \( 2 \times 1350 \times 750 = 2025000 \)
4. \( \cos(140^\circ) \approx -0.7660 \)

Plugging in the values:
\[
d = \sqrt{1822500 + 562500 - 2025000 \times (-0.7660)}
\]
\[
d = \sqrt{1822500 + 562500 + 1558650} 
\]
\[
d = \sqrt{3943650} \approx 1986 \, \text{km}
\]

### Answer
After 5 hours, the planes are approximately **1986 km apart**.

Would you like further details on any steps, or have any questions?

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### Related Questions:
1. How does the Law of Cosines apply to solving vector distance problems?
2. What would be the separation if the planes had traveled in perpendicular directions?
3. How does changing one plane's speed affect the final distance?
4. If the planes flew for 10 hours instead of 5, how far apart would they be?
5. How do you find the angle between vectors in other coordinate systems?

### Tip:
When using trigonometric functions, ensure your calculator is set to the correct angle unit (degrees or radians) based on the problem requirements.

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 hr, how far apart are the planes?

Calculate the distance between two airplanes after 5 hours of flight in different directions using the Law of Cosines. This problem involves trigonometric concepts, including vector analysis and angle measurement. Suitable for high school students in Grades 10-12.

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