Question

Two poles of equal heights are standing opposite to each other on either side of the road, which is 120 feet wide. From a point between them on the road, the angles 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.

Answer 1

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 = (120 - x) metres.

It is given that

∠APB = 60°

and, ∠CPD = 30°

In Δ APB, we have

$\frac{AP}{AP}$ = tan 60°

⇒ $\frac{h}{x} = \sqrt{3}$

⇒ h = $\sqrt{3} \ x$            .............(1)

In Δ CPD, we have

$\frac{CD}{CP}$ = tan 30°

⇒ $\frac{h}{120 - <em>x</em>} = \frac{1}{\sqrt{3}}$

⇒ h = $\frac{120 - <em>x</em>}{\sqrt{3}}$          ...............(2)

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

$\sqrt{3} \ <em>x</em> =\frac{120 - x}{\sqrt{3}}$

⇒ 3x = 120 - x

⇒ 4x = 120

⇒ x = 30

Putting x = 30 in (1), we get

h = $\sqrt{3} \times 30 = (1.732) \times 30 = 51.96$

Thus, the required point is at a distance of 30 metres from the first pole and 90 metres from the second pole.

The height of the pole is 51.96 metres.