Question

Two poles of equal heights are standing opposite to 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 ${60}^o$ and ${30}^o$ respectively. Find the height of the poles and the distances of the point from the poles.

• A. $21\sqrt{11}m,60m$
• B. $15\sqrt{7}m,56m$
• C. $20\sqrt{3}m,60m$
  
• D. $15\sqrt{2}m,45m$

Answer 1

The correct option is C $20\sqrt{3}m,60m$

Consider the above figure
$\tan {60}^0=\frac{b}{80-x}$
$\Rightarrow \sqrt{3}=\frac{b}{80-x}$
$\Rightarrow b=\sqrt{3}x-80\sqrt{3}$ ....(i)
and,
$\tan {30}^0=\frac{1}{\sqrt{3}}=\frac{a}{x}$
$x=a\sqrt{3}$ ...(ii)
$b=3a-80\sqrt{3}$
Or
$3a-b=80\sqrt{3}$
Now it is given that the heights are equal.
Hence
$a=b=h$
Therefore
$2h=80\sqrt{3}$
$h=20\sqrt{3}$
$=2\left(17.32\right)$
$=34.64m$
Hence
$x=h\sqrt{3}$
$=20\left(3\right)=60m$