From a tower 126 m high, the angles of depression of two rocks which are in a horizontal line through the base of the tower are 16⁰ and 12⁰ 20’. Find the distance between the rocks if they are on

(i) the same side of the tower

(ii) the opposite sides of the tower.

Solution:

Consider CD as the tower of height = 126 m

A and B are the two rocks on the same line

Angles of depression are 16⁰ and 12⁰ 20’

In triangle CAD

tan θ = CD/AD

Substituting the values

tan 16⁰ = 126/x

0.2867 = 126/x

So we get

x = 126/0.2867

x = 439.48

In right triangle CBD

tan 12⁰ 20’ = 126/y

So we get

0.2186 = 126/y

y = 126/0.2186 = 576.40

(i) In the first case

On the same side of the tower

AB = BD – AD

AB = y – x

Substituting the values

AB = 576.40 – 439.48

AB = 136.92 m

(ii) In the second case

On the opposite side of the tower

AB = BD + AD

AB = y + x

Substituting the values

AB = 576.40 + 439.48

AB = 1015.88m