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 7400 feet. Fatoumata initially measures an angle of elevation of \( 17^{\circ} \) to the plane at point \( A \). At some later time, she measures an angle of elevation of \( 34^{\circ} \) 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 determine the distance the plane traveled from point \( A \) to point \( B \), we'll analyze the situation using trigonometry. **Given:** - The plane maintains a constant altitude of \( h = 7,\!400 \) feet. - Angle of elevation at point \( A \): \( \theta_A = 17^\circ \). - Angle of elevation at point \( B \): \( \theta_B = 34^\circ \). **Approach:** 1. **Determine Horizontal Distances:** - At point \( A \), the horizontal distance (\( x \)) from Fatoumata to the plane can be found using the tangent function: \[ \tan(\theta_A) = \frac{h}{x} \implies x = \frac{h}{\tan(\theta_A)} \] - Similarly, at point \( B \), the horizontal distance (\( x' \)) is: \[ \tan(\theta_B) = \frac{h}{x'} \implies x' = \frac{h}{\tan(\theta_B)} \] 2. **Calculate the Horizontal Distances:** - Compute \( \tan(17^\circ) \approx 0.305730 \) and \( \tan(34^\circ) \approx 0.674508 \). - Therefore: \[ x = \frac{7,\!400}{0.305730} \approx 24,\!204.36 \text{ feet} \] \[ x' = \frac{7,\!400}{0.674508} \approx 10,\!970.96 \text{ feet} \] 3. **Determine the Distance Traveled by the Plane:** - The plane travels the difference between these two horizontal distances: \[ \text{Distance} = x - x' \approx 24,\!204.36 - 10,\!970.96 \approx 13,\!233.4 \text{ feet} \] - Rounding to the nearest foot: \[ \text{Distance} \approx 13,\!233 \text{ feet} \] **Final Answer:** The plane traveled approximately **13,233 feet** from point \( A \) to point \( B \).