Question

A tower is a tall structure, taller than it is wide, often by a significant margin. Towers are

distinguished from masts by their lack of guy-wires and are therefore, along with tall apartment

buildings, self-supporting structures. Towers are specifically distinguished from "apartment

buildings" in that they are not built to be habitable but to serve other functions.

Karan went to city and he saw a transmission tower fixed at the top of a high building. He come

to know that the height of the building is 20m. From a point on the ground, the angles of

elevation of the top and the bottom of a transmission tower are  and  respectively such that

cos
$\alpha$
= sin(150

–
$\alpha$
) and sin2
$\beta$
= cos(135

– 3
$\beta$
). Find the height of the tower.
​

Answer 1

Karan went to city and he saw a transmission tower fixed at the top of a high building.

Given,

cos α = sin (150 - 2α)

⇒ cos α = cos [90 - (150 - 2α) ]

⇒ cos α = cos [90 - 150 + 2α ]

⇒ cos α = cos [-60 + 2α ]

α = -60 + 2α

α - 2α = -60

-α = -60

∴ α = 60°

sin 2β = cos (135 - 3β)

⇒ sin 2β = sin [90 -  (135 - 3β) ]

⇒ sin 2β = sin [90 -  135 + 3β) ]

⇒ sin 2β = sin [ -45 + 3β ]

2β =  -45 + 3β

2β - 3β = -45

-β = -45

∴ β = 45°

Consider the figure while going through the following steps:

tan α = (20 + h) / AB

AB = (20 + h) / tan α ............(1)

tan β = 20 / AB

AB = 20 / tan β ................(2)

using (1) and (2), we get,

(20 + h) / tan α = 20 / tan β

(20 + h) / 20 = tan α  / tan β

1 + h/20 = tan α  / tan β

h / 20 = tan α / tan β - 1

The height of the tower,

h / 20 = tan α / tan β - 1

h / 20 = ( tan 60° / tan 45° )  - 1

h / 20 = (√3/1)  - 1

h = 20 (√3 - 1) m

Therefore the height of the tower is 20 (√3 - 1) m