Question

kayla 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 6875 feet. kayla initially measures an angle of elevation of 16° to the plane at point a. at some later time, she measures an angle of elevation of 30° 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. answer attempt 2 out of 2

Explanation:

Step1: Find horizontal distance to point A

Let $x_A$ = horizontal distance from Kayla to point A. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so:
$\tan(16^\circ) = \frac{6875}{x_A}$
$x_A = \frac{6875}{\tan(16^\circ)}$
Calculate $\tan(16^\circ)\approx0.2867$, so $x_A\approx\frac{6875}{0.2867}\approx24009$ feet

Step2: Find horizontal distance to point B

Let $x_B$ = horizontal distance from Kayla to point B. Use the same tangent formula:
$\tan(30^\circ) = \frac{6875}{x_B}$
$x_B = \frac{6875}{\tan(30^\circ)}$
Calculate $\tan(30^\circ)=\frac{1}{\sqrt{3}}\approx0.5774$, so $x_B\approx\frac{6875}{0.5774}\approx11907$ feet

Step3: Calculate distance from A to B

The plane travels along the horizontal line, so distance $AB = x_A - x_B$
$AB\approx24009 - 11907 = 12102$ feet

Answer:

12102 feet