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°. If the bridge is at a height of 8 m from the banks, then find the width of the river.

Answer 1

Clearly, ∠ABC = 45° and ∠ADC = 30° ....[∵ Angle of depression = Angle of elevation]

Now, In ΔABC, we have

tan 45° = `("AC")/("BC")`

⇒ 1 = `8/("BC")`

⇒ BC = 8 m  ...(i)

Also, In ΔACD, we have

tan 30° = `("AC")/("DC")`

⇒ `1/sqrt(3) = 8/("DC")`

⇒ DC = `8sqrt(3)`  ...(ii)

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

BD = BC + DC

= `8 + 8sqrt(3)`

= `8(sqrt(3) + 1)`

= 8(1.732 + 1)

= 8 × 2.732

= 21.856 m