Q) 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 3 m from the banks, then find the width of the river.

Ans:

Step 1: Diagram for this question:

Step 2: Calculation for the bridge length on LHS:

Let ‘s start with Δ ARB,

tan ∠ ARB = 

∴ tan 45° =

∴ D₁ = 3 ……….. (i)

Step 3: Calculation for the bridge length on RHS:

Next in Δ BRQ,

tan ∠ BRQ = 

∴ tan 30° =

∴

∴ D₂  = 3√3 ….. (ii)

Step 4: Combining the results:

Since, Width of the river = Bridge length AB = AR + BR = D₁ + D₂

From equations (i) and (ii), we get:

AB  = 3 + 3√3 = 3 (√3 + 1) m 

 Therefore, the width of the river is 3(√3 + 1) m.