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.