Question

lucy is trying to find the height of a radio antenna on the roof of a local building. she stands at a horizontal distance of 21 meters from the building. the angle of elevation from her eyes to the roof (point a) is 21°, and the angle of elevation from her eyes to the top of the antenna (point b) is 36°. if her eyes are 1.58 meters from the ground, find the height of the antenna (the distance from point a to point b). round your answer to the nearest meter if necessary.

Explanation:

Step1: Find the height from Lucy's eyes to point A

Let \(h_1\) be the height from Lucy's eyes to point A.
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here \(\theta = 21^{\circ}\) and adjacent side \(x = 21\) meters.
So \(h_1=21\times\tan(21^{\circ})\)
Using a calculator, \(\tan(21^{\circ})\approx0.3839\), then \(h_1 = 21\times0.3839=8.0619\) meters.

Step2: Find the height from Lucy's eyes to point B

Let \(h_2\) be the height from Lucy's eyes to point B.
Here \(\theta = 36^{\circ}\) and adjacent side \(x = 21\) meters.
So \(h_2=21\times\tan(36^{\circ})\)
Using a calculator, \(\tan(36^{\circ})\approx0.7265\), then \(h_2=21\times0.7265 = 15.2565\) meters.

Step3: Calculate the height of the antenna (AB)

The height of the antenna \(AB=h_2 - h_1\)
\(AB=15.2565-8.0619 = 7.1946\) meters.

Answer:

The height of the antenna is approximately \(7\) meters.