Question

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

Answer 1

Let the height of the poles be $h$

Then $AB=ED=h$

$BD=80m$ [given]

Let the given point be $C$ such that $BC=x$ and $CD=80-x$

In $△ABC,$

$\tan {30}^{\circ }=\frac{AB}{BC}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x}$

$\Rightarrow x=\sqrt{3}h$

In $△ECD,$

$\tan {60}^0=\frac{ED}{CD}$

$\Rightarrow \sqrt{3}=\frac{h}{80-x}$

$\Rightarrow h=(80-x)\sqrt{3}$

Put the value $x=\sqrt{3}h$

we get $h=(80-\sqrt{3}h)\sqrt{3}$

$\Rightarrow h=80\sqrt{3}-3h$

$\Rightarrow 4h=80\sqrt{3}$

$\Rightarrow h=20\sqrt{3}$

Then, height of poles $h=20\sqrt{3}m$