Question

Two poles of equal heights are standing opposite each other on either side of the road, which is $80m$ wide. From a point between them on the road, the angles of elevation of the top of the poles are ${60}^{\circ }$ and ${30}^{\circ }$, respectively. Find the height of the poles and the distances of the point from the poles.

Answer 1

Let $AB$ and $DE$ be the poles and $BD$ be the road,

In $△ABC$,

$\tan 60=\frac{AB}{BC}$

$BC=\frac{AB}{\sqrt{3}}$

$AB=\sqrt{3}BC$...(i)

In $△DEC$,

$\tan 30=\frac{DE}{CD}$

$\Rightarrow \frac{DE}{80-BC}=\frac{1}{\sqrt{3}}$

$AB=DE$...(poles of same height)

$\Rightarrow \frac{\sqrt{3}BC}{80-BC}=\frac{1}{\sqrt{3}}$ [From (i)]

$\Rightarrow 3BC=80-BC$

$\Rightarrow 4BC=80$

$\Rightarrow BC=20m$

Therefore, height of the poles $=20\sqrt{3}m$

Distance of the poles are $20m$ and $80-20=60m$ respectively.