Question

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.

Answer 1

Let AB and CD be the poles and O is the point from where the elevation angles are measured.

In ∆ABO,

$\frac{AB}{BO} = tan 60°$

$\frac{AB}{BO} = \sqrt3$

$BO = \frac{AB}{ \sqrt3}$

In ∆CDO,

$\frac{CD}{ DO} = tan 30°$

$\frac{CD }{ 80- BO} =\frac{ 1}{ \sqrt3 }$

$CD \sqrt3 = 80 -BO$

$CD\sqrt3 = 80 - \frac{AB}{ \sqrt3}$

$CD \sqrt3 + \frac{AB}{\sqrt3} = 80$

Since the poles are of equal heights,

$CD = AB$

$CD [\sqrt3 + \frac{1}{ \sqrt3}\, ] = 80$

$CD (\frac{3 +1}{ \sqrt3}) = 80$

$CD = 20\sqrt3 m$

$BO = \frac{AB}{ \sqrt3} = \frac{CD}{\sqrt3} = (\frac{20 \sqrt3}{\sqrt3} )m = 20m$

$DO = BD − BO = (80 − 20) m = 60 m$

Therefore, the height of poles is $20\sqrt3 m$ and the point is 20 m and 60 m far from these poles.