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 angle of elevation of the top of the poles are 60º and 30º respectively. Find the height of the poles and the distances of the point from the poles

Let AB and CD be two poles each of height h metres. Let P be a point on the road such that AP = x metres. Then CP = (80 – x) metres. Its is given that \( ∠ \ APB \ = \ 60º \) and \( ∠ \ CPD \ = \ 30º \).



In \( ∆ \ APB \), we have

\( \frac{AB}{AP} \ = \ tan60º \ \)
\( => \ \frac{h}{x} \ = \ \sqrt{3} \)
\( => \ h \ = \ \sqrt{3}x \) (1)

In \( ∆ \ CPD \), we have

\( \frac{CD}{CP} \ = \ tan30º \ \)
\( => \ \frac{h}{80-x} \ = \ \frac{1}{ \sqrt{3}} \)
\( => \ h \ = \ \frac{80-x}{ \sqrt{3}} \) (2)

Equating the values of h from (1) and (2), we get

\( \sqrt{3}x \ = \ \frac{80-x}{ \sqrt{3}} \ \)
\( =>\ 3x \ = \ 80-x \)
\( => \ x \ = \ 20 \)

Putting x = 20 in (1), we get

\( h \ = \ \sqrt{3} × 20 \ \)
\( = \ (1.732) × 20 \ = \ 34.64 \)

\(\therefore \) The point is at a distance of 20 metres from the first pole and 60 metres from the second pole. And the height of the pole is 34.64 metres.