Question

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are ${30}^{\circ }$ and ${45}^{\circ }$, respectively. If the bridge is at a height of 6 m from the banks, find the width of the river.

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Answer 1

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.

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$\angle PAD=\angle ADC\text{and}\angle QAB=\angle ABC\text{(Alternate angles)}$

$\text{tan}\angle ADC=\frac{AC}{DC}=\text{tan}{45}^{\circ }\Rightarrow \text{tan}{45}^{\circ }=\frac{6}{DC}\Rightarrow DC=6m\text{Also, tan}\angle ABC=\frac{AC}{BC}$ \\
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$\Rightarrow BC=\frac{AC}{\text{tan}\angle ABC}=\frac{6}{\text{tan}{30}^{\circ }}=6\sqrt{3}m$

$\therefore \text{Width of river}=BC+DC=6+6\sqrt{3}=6(\sqrt{3}+1)m$
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