A Cessna 150 and a Bombardier Dash 8 leave Calgary Airport in different directions.  The Cessna travels at 196 km/h, and the Dash 8 travels at 667 km/h.  After 2.5 hours, the two aircraft are 1,400 km apart.  What is the angle between their flight paths?  Answer in degrees, and round your answer to one decimal place.

To determine the angle between the flight paths of the Cessna 150 and the Bombardier Dash 8, we can use the Law of Cosines. 

First, calculate the distances each aircraft has traveled after 2.5 hours:
- Cessna 150 distance: \( 196 \text{ km/h} \times 2.5 \text{ h} = 490 \text{ km} \)
- Bombardier Dash 8 distance: \( 667 \text{ km/h} \times 2.5 \text{ h} = 1667 \text{ km} \)

Let \( a = 490 \) km, \( b = 1667 \) km, and \( c = 1400 \) km (the distance between the two aircraft).

The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cos(\theta)
\]

Plug in the known values:
\[
1400^2 = 490^2 + 1667^2 - 2 \times 490 \times 1667 \cos(\theta)
\]

Calculate the squares:
\[
1400^2 = 1960000
\]
\[
490^2 = 240100
\]
\[
1667^2 = 2780889
\]

Substitute these into the equation:
\[
1960000 = 240100 + 2780889 - 2 \times 490 \times 1667 \cos(\theta)
\]

Simplify:
\[
1960000 = 3020989 - 2 \times 490 \times 1667 \cos(\theta)
\]

Calculate the product:
\[
2 \times 490 \times 1667 = 1639660
\]

Substitute and solve for \( \cos(\theta) \):
\[
1960000 = 3020989 - 1639660 \cos(\theta)
\]
\[
1960000 - 3020989 = -1639660 \cos(\theta)
\]
\[
-1060989 = -1639660 \cos(\theta)
\]
\[
\cos(\theta) = \frac{1060989}{1639660}
\]
\[
\cos(\theta) \approx 0.6471
\]

Finally, find \( \theta \) by taking the arccosine:
\[
\theta = \arccos(0.6471) \approx 49.4^\circ
\]

Thus, the angle between their flight paths is approximately \( 49.4^\circ \).

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What is the Law of Cosines and how is it applied?
2. How do you derive the Law of Cosines?
3. What other methods can be used to find the angle between two paths if their distances and separation are known?
4. How can you calculate distances in 3-dimensional space using vectors?
5. How do relative speeds affect the calculation of angles between moving objects?
6. What are some practical applications of the Law of Cosines in navigation?
7. Can the Law of Sines be used in this scenario? If so, how?
8. How can we verify the results obtained using the Law of Cosines?

**Tip:** When dealing with angle problems in navigation or physics, breaking down the vectors and using trigonometric identities can often simplify complex problems.

Learn how to calculate the angle between the flight paths of two aircraft using the Law of Cosines. Detailed step-by-step solution with calculations provided. Suitable for high school students studying trigonometry and geometry.

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