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# Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height

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

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Given that,
The height of both poles are equal  DC = AB. The angle of elevation of of the top of the poles are $\angle DEC=30^o$ and $\angle AEB=60^o$ resp.
Let the height of the poles be  $h$ m and CE = $x$ and BE = 80 - $x$

According to question,
In triangle DEC,

$\\\Rightarrow \tan 30^o = \frac{DC}{CE} = \frac{h}{x}\\\\\Rightarrow \frac{1}{\sqrt{3}}= \frac{h}{x}\\\\\Rightarrow x=h\sqrt{3}$..............(i)

In triangle AEB,
$\\\Rightarrow \tan 60^o = \frac{AB}{BE}=\frac{h}{80-x}\\\\\Rightarrow \sqrt{3}=\frac{h}{80-x}\\\\\Rightarrow x=80 - \frac{h}{\sqrt{3}}$..................(ii)
On equating eq (i) and eq (ii), we get

$\sqrt{3}h=80 - \frac{h}{\sqrt{3}}$
$\frac{h}{\sqrt{3}}=20$
$h=20\sqrt{3}$ m
So, $x$ = 60 m

Hence the height of both poles is ($h=20\sqrt{3}$)m and the position of the point is at 60 m from the pole CD and 20 m from the pole AB.

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