Question

From a window (60 m high above the ground) of a house in a street the angles of elevation and depression of the top and the foot of another house on opposite side of street are 60° and 45° respectively. Show that the height of the opposite house is `60(1 + sqrt(3))` m.

Answer 1

Given: From a window 60 m above the ground, the angle of elevation of the top of an opposite house is 60° and the angle of depression of its foot is 45°.

Step-wise calculation:

1. Let the horizontal distance between the window and the foot of the opposite house be x and let the height of the opposite house be H.

2. Angle of depression to the foot = 45°

⇒ `tan 45^circ = "Vertical drop"/"Horizontal distance"`

= `60/x`

Since tan 45° = 1,

`1 = 60/x` 

⇒ x = 60

3. Angle of elevation to the top = 60°

⇒ `tan 60^circ = "Height difference"/"Horizontal distance"` 

= `(H - 60)/x` 

Since `tan 60^circ = sqrt(3)` and x = 60,

`sqrt(3) = (H - 60)/60`

⇒ `H - 60 = 60sqrt(3)`

4. Solve for H:

`H = 60 + 60sqrt(3)` 

`H = 60(1 + sqrt(3))`

The height of the opposite house is `60(1 + sqrt(3))` m.