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 2.5 m from the banks, find the width of the river. [Take `sqrt(3) = 1.732`.]

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^circ = 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^circ = 1`

`⇒ 2.5/(BD) = 1`

⇒ BD = 2.5 m 

∴ Width of the river = AB

= (AD + BD)

= `(2.5sqrt(3) + 2.5)`

= 6.83 m