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Anna is trying to find the height of a radio antenna on the roof of a local building. Shestands at a horizontal distance of 19 meters from the building. The angle of elevationfrom her eyes to the roof (point A) is 34°, and the angle of elevation from her eyes tothe top of the antenna (point B) is 41°. If her eyes are 1.5 meters from the ground,find the height of the antenna (the distance from point A to point B). Round youranswer to the nearest tenth of a meter if necessary.

Sagot :

SOLUTION

Step 1: Let us draw a pictorial view of the information given:

The height of the antenna AB will be:

[tex]AB=OB-OA[/tex]

Let us find OA from triangle OAS

[tex]\begin{gathered} \tan 34=\frac{OA}{19} \\ OA=19\times\tan 34 \\ OA=12.816m \end{gathered}[/tex]

Now, we will find OB from triangle OBS

[tex]\begin{gathered} \tan 41=\frac{OB}{19} \\ OB=19\times\tan 41 \\ OB=16.516m \end{gathered}[/tex]

Now, let us find length AB, which is the length of the antenna:

[tex]\begin{gathered} AB=OB-OA \\ AB=16.516m-12.816m \\ AB=3.7m \end{gathered}[/tex]

The height of the antenna is 3.7 meters.

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