A radio antenna

Avanti 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 42°, and the angle of elevation from her eyes to the top of the antenna (point B) is 51°. If her eyes are 1.54 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.

Final Answer:

h =  7 m

Step-by-step explanation:

$$\begin{gathered} x=21\text{m} \\ \alpha =42{}^{\circ } \\ \beta =51{}^{\circ } \\ {h}_1=1.54\text{m} \\ \tan \alpha =y_1:x \\ {y}_1=x\cdot \tan \alpha =x\cdot \tan 42°=21\cdot \tan 42°=21\cdot 0.900404=18.90848\text{m} \\ {H}_1=y_1+h_1=18.9085+1.54\doteq 20.4485\text{m} \\ \tan \beta =y_2:x \\ {y}_2=x\cdot \tan \beta =x\cdot \tan 51°=21\cdot \tan 51°=21\cdot 1.234897=25.93284\text{m} \\ h=y_2-y_1=25.9328-18.9085=7\text{m} \end{gathered}$$

Try another example