Question

amira 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 6650 feet. amira 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 35° to the plane at point b. find the distance the plane traveled from point a to point b. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Find horizontal distances

Let the horizontal distance from the point directly below the plane to point \(A\) be \(x_1\), and to point \(B\) be \(x_2\).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For the angle of elevation of \(17^{\circ}\) (at point \(A\)): \(\tan17^{\circ}=\frac{6650}{x_1}\), so \(x_1 = \frac{6650}{\tan17^{\circ}}\).
Using a calculator, \(\tan17^{\circ}\approx0.3057\), then \(x_1=\frac{6650}{0.3057}\approx21753.4\).
For the angle of elevation of \(35^{\circ}\) (at point \(B\)): \(\tan35^{\circ}=\frac{6650}{x_2}\), so \(x_2=\frac{6650}{\tan35^{\circ}}\).
Using a calculator, \(\tan35^{\circ}\approx0.7002\), then \(x_2 = \frac{6650}{0.7002}\approx9497.3\).

Step2: Calculate the distance from \(A\) to \(B\)

The distance from point \(A\) to point \(B\) is \(d=x_1 - x_2\).
\(d=21753.4−9497.3 = 12256.1\)

Answer:

\(12256.1\)