Question

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at a height of 2.5m from the banks, find the width of the river.

Answer 1

Let A and B be two points on the banks on the opposite side of the river and P be the point on the bridge at a height of 2.5 m.

Thus, we have:
DP = 2.5, ∠PAD = 30°  and ∠PBD = 45°
In the right  ΔAPD,we have:

`(DP)/(AD) = tan 30° = 1/sqrt(3)`

`⇒ 2.5 /(AD) = 1/ sqrt(3)`

`⇒ AD = 2.5 sqrt(3) m`

In the right   ΔPDB,we have:

`(DP)/(BD) = tan 45° = 1`

`⇒ 2.5/(BD) = 1`

⇒ BD=2.5m 

`∴"Width of the river" = AB = ( AD+ BD) =(2.5sqrt(3) + 2.5) = 6.83m`