Question

cai 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 6550 feet. cai initially measures an angle of elevation of 16° to the plane at point a. at some later time, he 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 distance at A

Let $x_A$ = horizontal distance from Cai to point below plane at A. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so:
$\tan(16^\circ)=\frac{6550}{x_A}$
$x_A=\frac{6550}{\tan(16^\circ)}$
$x_A\approx\frac{6550}{0.2867}\approx22846.2$ feet

Step2: Find horizontal distance at B

Let $x_B$ = horizontal distance from Cai to point below plane at B.
$\tan(35^\circ)=\frac{6550}{x_B}$
$x_B=\frac{6550}{\tan(35^\circ)}$
$x_B\approx\frac{6550}{0.7002}\approx9354.5$ feet

Step3: Calculate distance AB

Distance AB = $x_A - x_B$
$AB\approx22846.2 - 9354.5$

Answer:

13491.7 feet