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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.
Solution
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
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