Question

question
cameron 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 6600 feet. cameron initially measures an angle of elevation of 18° to the plane at point a. at some later time, he measures an angle of elevation of 37° 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 1 out of 2
feet
submit answer
show examples

Explanation:

Step1: Find the initial horizontal distance

Let the initial horizontal distance from Cameron to the point directly below the plane (at point \(A\)) be \(x_1\).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Given the altitude \(h = 6600\) feet and \(\theta_1=18^{\circ}\), and \(\tan(18^{\circ})=\frac{h}{x_1}\).
So, \(x_1=\frac{h}{\tan(18^{\circ})}=\frac{6600}{\tan(18^{\circ})}\).
Using a calculator, \(\tan(18^{\circ})\approx0.3249\), then \(x_1=\frac{6600}{0.3249}\approx 20314\) feet.

Step2: Find the later horizontal distance

Let the later horizontal distance from Cameron to the point directly below the plane (at point \(B\)) be \(x_2\).
Given \(\theta_2 = 37^{\circ}\) and \(h = 6600\) feet, and \(\tan(37^{\circ})=\frac{h}{x_2}\).
So, \(x_2=\frac{h}{\tan(37^{\circ})}\).
Using a calculator, \(\tan(37^{\circ})\approx0.7536\), then \(x_2=\frac{6600}{0.7536}\approx8758\) feet.

Step3: Calculate the distance the plane traveled

The distance \(d\) the plane traveled from point \(A\) to point \(B\) is \(d=x_1 - x_2\).
\(d=20314−8758 = 11556\) feet.

Answer:

\(11556\)