Fatoumata spots an airplane on radar that is currently approaching in a straight line, and that will fly directly overhead. The plane maintains a constant altitude of 6875 feet. Fatoumata initially measures an angle of elevation of 17 degrees to the plane at point A. At some later time, she measures an angle of elevation of 40 degrees to the plane at point B. Find the distance the plane traveled from point A to point B. Round your answer to the nearest foot if necessary.

To solve this problem, we can use trigonometry, particularly the tangent function, which relates the angle of elevation, the altitude of the airplane, and the horizontal distance from the observer to the point directly beneath the airplane.

### Step 1: Set up the problem
- Let \( h = 6875 \) feet be the altitude of the plane.
- Let \( d_1 \) be the horizontal distance from point A to the point directly under the airplane when the angle of elevation is 17 degrees.
- Let \( d_2 \) be the horizontal distance from point B to the point directly under the airplane when the angle of elevation is 40 degrees.
- Let \( x \) be the distance the plane traveled from point A to point B.

We need to find \( x = d_1 - d_2 \).

### Step 2: Use the tangent function
For point A:
\[
\tan(17^\circ) = \frac{h}{d_1}
\]
\[
d_1 = \frac{h}{\tan(17^\circ)} = \frac{6875}{\tan(17^\circ)}
\]

For point B:
\[
\tan(40^\circ) = \frac{h}{d_2}
\]
\[
d_2 = \frac{h}{\tan(40^\circ)} = \frac{6875}{\tan(40^\circ)}
\]

### Step 3: Calculate the values of \( d_1 \) and \( d_2 \)
First, find the values of \( \tan(17^\circ) \) and \( \tan(40^\circ) \):
\[
\tan(17^\circ) \approx 0.3057, \quad \tan(40^\circ) \approx 0.8391
\]

Now, calculate \( d_1 \) and \( d_2 \):
\[
d_1 = \frac{6875}{0.3057} \approx 22491 \text{ feet}
\]
\[
d_2 = \frac{6875}{0.8391} \approx 8193 \text{ feet}
\]

### Step 4: Find the distance the plane traveled
\[
x = d_1 - d_2 = 22491 - 8193 \approx 14298 \text{ feet}
\]

### Final Answer
The distance the plane traveled from point A to point B is approximately **14,298 feet**.

Would you like more details or have any questions?

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**Related Questions:**
1. How would the distance change if the plane’s altitude were different?
2. What if the angles of elevation were measured in radians instead of degrees?
3. How would you calculate the distance traveled if the angle of elevation at point B were less than at point A?
4. Can you derive a general formula for the distance traveled based on the angles and altitude?
5. What happens to the distance if one of the angles of elevation is 90 degrees?

**Tip:** In problems involving angles and distances, drawing a diagram often helps in understanding the relationships between different components.

Learn how to calculate the distance traveled by a plane using angles of elevation. Solve a trigonometry problem involving constant altitude and varying angles of elevation. Suitable for high school students studying trigonometry and geometry.

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