Question

cai is trying to find the height of a radio antenna on the roof of a local building. he stands at a horizontal distance of 21 meters from the building. the angle of elevation from his eyes to the roof (point a) is 24°, and the angle of elevation from his eyes to the top of the antenna (point b) is 35°. if his 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 tenth of a meter if necessary. answer attempt 1 out of 2

Explanation:

Step1: Find the height from eye - level to point A

We use the tangent function for right - triangles. $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For the angle of elevation to point A ($\theta = 24^{\circ}$) and adjacent side $x = 21$ meters. Let $h_1$ be the height from eye - level to point A. So, $\tan(24^{\circ})=\frac{h_1}{21}$, and $h_1 = 21\times\tan(24^{\circ})$.
$h_1=21\times0.4452\approx9.35$ meters.

Step2: Find the height from eye - level to point B

For the angle of elevation to point B ($\theta = 35^{\circ}$) and adjacent side $x = 21$ meters. Let $h_2$ be the height from eye - level to point B. Using the tangent function $\tan(35^{\circ})=\frac{h_2}{21}$, and $h_2 = 21\times\tan(35^{\circ})$.
$h_2=21\times0.7002\approx14.70$ meters.

Step3: Find the height of the antenna (from A to B)

The height of the antenna $h=h_2 - h_1$.
$h=14.70 - 9.35=5.35\approx5.4$ meters.

Answer:

$5.4$ meters