Question

Two poles of equal heights are standing opposite to each other on either side of the road whichch is $80m$ wide. From a point $P$ between them on the road, the angle of elevation of the top of a pole is ${60}^o$ and the angle of depression from the top of another pole at point $P$ is ${30}^o$.Find the heights of the poles and the distances of the point $P$ from the poles?

Answer 1

Let $AB$ and $CD$ be the two poles of equal height.

Their heights be $h$ metres.

$BC$ be the $80m$ wide road.

$P$ be any point on the road.

Let ,$CP$ be $x$ metres.

Then, $BP=(80–x).$

Also, $\angle APB=60°$ and $\angle DPC=30°$

It can be observed that $△PAB$ and $△PDC$ are right angled triangles.

In $△DCP,$

$\tan {30}^o=\frac{CD}{CP}$

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

$\Rightarrow h=\frac{x}{\sqrt{3}}$ ---------- $\left(1\right)$

In $△ABP$

$\tan 60°=\frac{AB}{BP}$

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

$\Rightarrow h=\sqrt{3}(80–x)$

$\Rightarrow \frac{x}{\sqrt{3}}=\sqrt{3}(80–x)$

$\Rightarrow x=3(80–x)$

$\Rightarrow x=240–3x$

$\Rightarrow x+3x=240$

$\Rightarrow 4x=240$

$\Rightarrow x=60$

Height of the pole, $h=\frac{x}{\sqrt{3}}=\frac{60}{\sqrt{3}}=20\sqrt{3}.$

Thus, position of the point $P$ is $60m$ from $C$ and height of each pole is $20\sqrt{3}m.$