We have a question from trigonometry of height and distance and it was already asked in previous year paper of 2015 in which we have been asked to find the height of each pole and the distance of a point from the poles, if Two poles of equal height are standing opposite to each other on the either side of the road which is 80 m wide. From a point P b/w them on road , the angle of elevation of the top of a pole is 60°and the angle of depression from the top of another pole at point P is 30°

Book RS Aggarwal, Class 10, Chapter 14, question no 11

AB and CD be the two poles of equal height.

Their heights be Hm.

BC be the 80m wide road.

P be any point on the road.

Let ,

CPbexm,

BP=(80–x).

Also, ∠APB=60° and ∠DPC=30°

In right angled triangle DCP,

Tan30°=CD/CP

⇒h/x=1/√3

⇒h=x/√3 ———- (1)

In right angled triangle ABP

Tan60°=AB/AP

⇒h/(80–x)=√3

⇒h=√3(80–x)

⇒x/√3=√3(80–x)

⇒x=3(80–x)

⇒x=240–3x

⇒x+3x=240

⇒4x=240

⇒x=60

Height of the pole, h=x/√3=60/√3=20√3.

Thus, position of the point P is 60m from C and height of each pole is 20√3m.