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 point A

Let $x_A$ be the horizontal distance from Cai to the point directly below the plane at point A. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so:
$$x_A = \frac{6550}{\tan(16^\circ)}$$
Calculate $\tan(16^\circ)\approx0.2867$, so $x_A\approx\frac{6550}{0.2867}\approx22846.2$ feet.

Step2: Find horizontal distance at point B

Let $x_B$ be the horizontal distance from Cai to the point directly below the plane at point B. Use the same tangent formula:
$$x_B = \frac{6550}{\tan(35^\circ)}$$
Calculate $\tan(35^\circ)\approx0.7002$, so $x_B\approx\frac{6550}{0.7002}\approx9354.5$ feet.

Step3: Calculate plane's travel distance

The distance the plane travels is the difference between the two horizontal distances:
$$\text{Distance} = x_A - x_B$$

Answer:

$\approx22846.2 - 9354.5 = 13491.7$ feet