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 3 m from the banks, find the width of the river. (Use `sqrt(3)` = 1.73)

Answer 1

Let the width of the river be x m

i.e., AB = x m

P is the point on the bridge

∴ PQ = 3 m

and ∠RPA = 30° and ∠SPB = 45°

In ΔAPQ, ∠Q = 90° ∠A = 30°

tan A = `(PQ)/(AQ)`

tan 30° = `3/(AQ)`

`1/sqrt(3) = 3/(AQ)`

∴ AQ = `3sqrt(3)` m  ...(1)

In ΔPQB, ∠Q = 90° ∠B = 45°

tan B = `(PQ)/(BQ)`

tan 45° = `3/(BQ)`

1 = `3/(BQ)`

BQ = 3  ...(2)

AB = AQ + BQ

= `3sqrt(3) + 3`

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

Width of River = `3(sqrt(3) + 1)`

= 3 × 2.73

= 8.19 m.