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 6 m from the banks, find the width of the river.
The situation can be represented by the figure below. PQ is the bridge and BD is the river with B and D as two banks.
∠PAD=∠ADC and ∠QAB=∠ABC (Alternate angles)
tan ∠ADC=ACDC= tan 45∘⇒tan 45∘=6DC⇒DC=6 mAlso, tan ∠ABC=ACBC
⇒BC=ACtan ∠ABC=6tan 30∘=6√3 m
∴Width of river=BC+DC=6+6√3=6(√3+1) m