A Boeing 707 makes the same trip as the Airbus in the previous question, except that the Boeing makes the flight in a straight line from origin to destination.  What is the bearing of the Boeing 707 when it departs?  Answer in degrees, and round your answer to one decimal place.

To determine the bearing of the Boeing 707, we need to find the direction it must take to travel directly from the origin to the destination, using the information given in the previous problem. This requires us to find the angle relative to one of the initial paths (e.g., the path of the Cessna 150) or the direction directly between the two points.

From the previous problem:
- The distance traveled by the Cessna 150 (\(A\)) is 490 km.
- The distance traveled by the Bombardier Dash 8 (\(B\)) is 1667 km.
- The distance between the two aircraft after 2.5 hours (\(C\)) is 1400 km.
- The angle between their paths is approximately 49.4 degrees.

We need to use this angle to determine the bearing of the Boeing 707.

The bearing of the Boeing 707 would be the direction from the origin to the destination directly. To find the bearing, we can use trigonometry. Let's consider the angle between the paths and the distances. We will use the Law of Sines to find the angle relative to one of the original paths.

We have:
\[
\frac{\sin(\theta)}{a} = \frac{\sin(A)}{c}
\]

Using the known distances:
\[
\frac{\sin(\theta)}{1667} = \frac{\sin(49.4^\circ)}{1400}
\]

First, calculate \(\sin(49.4^\circ)\):
\[
\sin(49.4^\circ) \approx 0.7587
\]

Now, solve for \(\sin(\theta)\):
\[
\sin(\theta) = \frac{1667 \times 0.7587}{1400} \approx 0.9032
\]

Find the angle:
\[
\theta = \arcsin(0.9032) \approx 64.5^\circ
\]

Since the bearing is the angle relative to the north (assuming standard navigation practice where bearings are measured clockwise from the north), and knowing the paths form a certain angle, the bearing would be the complement of the found angle relative to the direction of travel of the Bombardier Dash 8.

Therefore, the bearing of the Boeing 707 when it departs is:
\[
\text{Bearing} = 360^\circ - 64.5^\circ = 295.5^\circ
\]

So the bearing is approximately \(295.5^\circ\).

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

1. How do you calculate bearings using trigonometry?
2. What is the significance of bearings in navigation?
3. How can you convert between different angular measurement systems (degrees, radians)?
4. How does the Earth's curvature affect long-distance navigation calculations?
5. What tools do pilots use to determine bearings?
6. Can bearings change during flight? If so, how is it accounted for?
7. How do you solve navigation problems using vector components?
8. What are the differences between true north, magnetic north, and grid north?

**Tip:** When dealing with navigation problems, always check the reference direction for your bearings (e.g., true north vs. magnetic north) to ensure accuracy in real-world applications.

Learn how to calculate the bearing of a Boeing 707 flying directly from origin to destination using trigonometry and the Law of Sines. The problem involves finding the angle relative to the paths of other aircraft and converting it into a bearing measurement. Suitable for high school students studying navigation and trigonometry.

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