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The problem described involves two airplanes leaving an airport at the same time. After an hour, they are traveling at different distances: one at 295 km, the other at 156 km, and the distance between them is 178 km. The task is to find the angle, \(\theta\), between their flight paths using the law of cosines.

### Law of Cosines Formula:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\]
Where:
- \(a = 295 \, \text{km}\)
- \(b = 156 \, \text{km}\)
- \(c = 178 \, \text{km}\)

We need to solve for \(\theta\). Rearranging the formula:
\[
\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Substituting the values:
\[
\cos(\theta) = \frac{295^2 + 156^2 - 178^2}{2 \cdot 295 \cdot 156}
\]
\[
\cos(\theta) = \frac{87025 + 24336 - 31684}{91920}
\]
\[
\cos(\theta) = \frac{79677}{91920} \approx 0.8669
\]

Finally, to find \(\theta\):
\[
\theta = \cos^{-1}(0.8669) \approx 29.9^\circ
\]

### Answer:
The angle between the two flight paths is approximately **29.9 degrees**.

Would you like more details or have any questions?

Here are some related questions to consider:
1. What happens to the angle if the distances of the planes change?
2. How can the law of sines be applied in similar problems?
3. How does rounding affect precision in trigonometric problems?
4. Can this problem be solved using a different method, like vectors?
5. How would the formula change if the problem was in three dimensions?

**Tip:** Always carry at least four decimal places in intermediate steps to minimize rounding errors when solving trigonometric problems.

Two airplanes leave an airport at the same time. One travels 295 km, the other 156 km, and the distance between them is 178 km. Find the angle between their flight paths using the law of cosines.

Learn how to calculate the angle between two airplanes' flight paths using the law of cosines. This problem involves trigonometry and geometric concepts, and we provide a detailed step-by-step solution for students in Grades 10-12.

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