Question

From the top of tower 60 m high, the angles of depression of the top and bottom of a pole are observed to be 45° and 60° respectively. Find the height of the pole.

Answer 1

Given:

Tower height AB = 60 m.

Angles of depression from the top of the tower to the top and bottom of the pole are 45° and 60° respectively.

Step-wise calculation:

1. Let the pole height = h (m) and the horizontal distance between the tower and the pole = d (m).

2. From the angle of depression 45° to the top of the pole:

`tan 45^circ = (60 - h)/d`

⇒ `1 = (60 - h)/d`

⇒ d = 60 – h

3. From the angle of depression 60° to the bottom (ground) of the pole:

`tan 60^circ = 60/d`

⇒ `sqrt(3) = 60/d`

⇒ `d = 60/sqrt(3)`

⇒ `d = 20sqrt(3)`

4. Equate the two expressions for d:

`60 - h = 20sqrt(3)` 

⇒ `h = 60 - 20sqrt(3)`

5. Numerical value:

`20sqrt(3) ≈ 34.64`

So, h ≈ 60 – 34.64 = 25.36 m.

Height of the pole = `60 − 20sqrt(3)` metres ≈ 25.36 metres.