Question

kadeesha 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 6150 feet. kadeesha initially measures an angle of elevation of 17° to the plane at point a. at some later time, she measures an angle of elevation of 33° 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.

Explanation:

Step1: Find horizontal distances

Let the horizontal distance from the point directly below the plane to point \(A\) be \(x_A\), and to point \(B\) be \(x_B\).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For the angle of elevation of \(17^{\circ}\) (at point \(A\)): \(\tan17^{\circ}=\frac{6150}{x_A}\), so \(x_A = \frac{6150}{\tan17^{\circ}}\).
Using a calculator, \(\tan17^{\circ}\approx0.3057\), then \(x_A=\frac{6150}{0.3057}\approx20118\).
For the angle of elevation of \(33^{\circ}\) (at point \(B\)): \(\tan33^{\circ}=\frac{6150}{x_B}\), so \(x_B=\frac{6150}{\tan33^{\circ}}\).
Using a calculator, \(\tan33^{\circ}\approx0.6494\), then \(x_B = \frac{6150}{0.6494}\approx9470\).

Step2: Calculate the distance traveled

The distance the plane traveled from point \(A\) to point \(B\) is \(d=x_A - x_B\).
Substitute the values of \(x_A\) and \(x_B\): \(d = 20118-9470=10648\).

Answer:

\(10648\) feet