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 $21^{\circ}$, and the angle of elevation from her eyes to the top of the antenna (point b) is $36^{\circ}$. 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: Define variables for heights

Let $h_A$ = height from eyes to roof, $h_B$ = height from eyes to antenna top, $d=21$ m (horizontal distance).

Step2: Calculate $h_A$ using tangent

$\tan(21^\circ)=\frac{h_A}{d}$
$h_A = d \times \tan(21^\circ) = 21 \times \tan(21^\circ)$
$h_A \approx 21 \times 0.3839 = 8.0619$ m

Step3: Calculate $h_B$ using tangent

$\tan(36^\circ)=\frac{h_B}{d}$
$h_B = d \times \tan(36^\circ) = 21 \times \tan(36^\circ)$
$h_B \approx 21 \times 0.7265 = 15.2565$ m

Step4: Find antenna height ($h_B - h_A$)

Antenna height = $15.2565 - 8.0619$

Answer:

8 meters (rounded to nearest meter)