Question

deepa is trying to find the height of a radio antenna on the roof of a local building. she stands at a horizontal distance of 27 meters from the building. the angle of elevation from her eyes to the roof (point a ) is 35°, and the angle of elevation from her eyes to the top of the antenna (point b) is 41°. if her eyes are 1.69 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: Calculate the height from her eyes to point A

We use the tangent function. Let \(h_1\) be the height from her eyes to point A.
\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), where \(\theta = 35^{\circ}\) and adjacent \(= 27\) meters.
\(h_1 = 27\times\tan(35^{\circ})\)
\(h_1=27\times0.7002\approx18.9054\) meters.

Step2: Calculate the height from her eyes to point B

Let \(h_2\) be the height from her eyes to point B. Using the tangent function with \(\theta = 41^{\circ}\) and adjacent \(= 27\) meters.
\(h_2 = 27\times\tan(41^{\circ})\)
\(h_2=27\times0.8693\approx23.4711\) meters.

Step3: Find the height of the antenna (AB)

\(AB=h_2 - h_1\)
\(AB=23.4711 - 18.9054=4.5657\approx5\) meters.

Answer:

5 meters.