Two poles of equal heights are standing opposite to each other on either side of the road which is $ 80 \mathrm{~m} $ 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.
Given:
Two poles of equal heights are standing opposite to each other on either side of the road which is \( 80 \mathrm{~m} \) 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.
To do:
We have to find the height of the poles and the distances of the point from the poles.
Solution:
Let $AB$ and $CD$ be the heights of the poles and $BD$ be the width of the road.
Let $O$ be the point of observation.
From the figure,
$\mathrm{BD}=80 \mathrm{~m}, \angle \mathrm{AOB}=60^{\circ}, \angle \mathrm{COD}=30^{\circ}$.
Let the heights of the poles be $\mathrm{AB}=\mathrm{CD}=h \mathrm{~m}$, the distance between the point $O$ and point $B$ be $\mathrm{BO}=x \mathrm{~m}$ and the distance between the point $O$ and point $D$ be $\mathrm{OD}=80-x \mathrm{~m}$.
We know that,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { AB }}{OB}$
$\Rightarrow \tan 60^{\circ}=\frac{h}{x}$
$\Rightarrow \sqrt3=\frac{h}{x}$
$\Rightarrow h=x\sqrt3 \mathrm{~m}$.........(i)
Similarly,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { CD }}{OD}$
$\Rightarrow \tan 30^{\circ}=\frac{h}{80-x}$
$\Rightarrow \frac{1}{\sqrt3}=\frac{h}{80-x}$
$\Rightarrow h=\frac{80-x}{\sqrt3} \mathrm{~m}$..........(ii)
From (i) and (ii), we get,
$\Rightarrow x\sqrt3=\frac{80-x}{\sqrt3} \mathrm{~m}$
$\Rightarrow (x\sqrt3)\sqrt3=80-x \mathrm{~m}$
$\Rightarrow 3x+x=80 \mathrm{~m}$
$\Rightarrow x=\frac{80}{4} \mathrm{~m}$
$\Rightarrow x=20 \mathrm{~m}$
$\Rightarrow 80-x=80-20=60 \mathrm{~m}$
$\Rightarrow h=20\sqrt3 \mathrm{~m}$
Therefore, the height of the poles is $20\sqrt3 \mathrm{~m}$, the distances of the point from the poles is $20 \mathrm{~m}$ and $60 \mathrm{~m}$ respectively.
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