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.

Question

A ladder 14 m long just reaches the top of a vertical wall. If the ladder makes an angle of $60^\circ$ with the wall, then the height of the wall is:

Answer 1

Two poles of equal heights are standing opposite each other on either side of the road,
Let AB and CD represent the poles, with O being the observation point.

In triangle ABO:

$\frac{AB}{BO} = tan 60°$

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

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

In triangle CDO:

$\frac{CD}{ DO} = tan 30°$

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

$CD \sqrt3 = 80 -BO $

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

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

Given that the poles have 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 $

The height of the poles is $20\sqrt3 m$. The observation point is located 20 m from one pole and 60 m from the other.

Solution and Explanation

Top Questions on Heights and Distances

Questions Asked in CBSE Class X exam