Question

rashaad 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 7075 feet. rashaad initially measures an angle of elevation of 20° to the plane at point a. at some later time, he measures an angle of elevation of 38° 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 distance to point A

Let $x_A$ be the horizontal distance from Rashaad to the point directly below A. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so:
$$\tan(20^\circ)=\frac{7075}{x_A} \implies x_A=\frac{7075}{\tan(20^\circ)}$$
Calculate $\tan(20^\circ)\approx0.3640$, so $x_A\approx\frac{7075}{0.3640}\approx19436.81$ feet.

Step2: Find horizontal distance to point B

Let $x_B$ be the horizontal distance from Rashaad to the point directly below B.
$$\tan(38^\circ)=\frac{7075}{x_B} \implies x_B=\frac{7075}{\tan(38^\circ)}$$
Calculate $\tan(38^\circ)\approx0.7813$, so $x_B\approx\frac{7075}{0.7813}\approx9055.42$ feet.

Step3: Calculate distance A to B

The distance between A and B is the difference in horizontal distances:
$$\text{Distance } AB = x_A - x_B$$
$$AB\approx19436.81 - 9055.42$$

Answer:

10381.4 feet