Question

christian is trying to find the height of a radio antenna on the roof of a local building. he stands at a horizontal distance of 16 meters from the building. the angle of elevation from his eyes to the roof (point a) is 26°, and the angle of elevation from his eyes to the top of the antenna (point b) is 46°. if his 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 tenth of a meter if necessary.

Explanation:

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

Use the tangent function \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For the angle of elevation to point \(A\) (\(\theta = 26^{\circ}\), adjacent \(=16\) meters). Let \(h_1\) be the height from eye - level to point \(A\).
\(\tan(26^{\circ})=\frac{h_1}{16}\), so \(h_1 = 16\times\tan(26^{\circ})\).
Using a calculator, \(\tan(26^{\circ})\approx0.4877\), then \(h_1=16\times0.4877 = 7.8032\) meters.

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

For the angle of elevation to point \(B\) (\(\theta = 46^{\circ}\), adjacent \( = 16\) meters). Let \(h_2\) be the height from eye - level to point \(B\).
\(\tan(46^{\circ})=\frac{h_2}{16}\), so \(h_2=16\times\tan(46^{\circ})\).
Using a calculator, \(\tan(46^{\circ})\approx1.0355\), then \(h_2 = 16\times1.0355=16.568\) meters.

Step3: Calculate the height of the antenna (\(AB\))

The height of the antenna \(AB=h_2 - h_1\).
\(AB=16.568 - 7.8032=8.7648\) meters.

Answer:

\(8.8\) meters