From a point on a bridge across the river, the angles of depressions of the banks on opposite sides of the river are $\displaystyle 30$° and $\displaystyle 60$° respectively. If the bridge is at a height of $\displaystyle 4$ m from the banks, find the width of the river.

Let AB be the width of river In right \(\displaystyle \Delta \mathrm{PAQ}\), \[\begin{aligned} & \tan 30^{\circ}=\frac{1}{\sqrt{3}}=\frac{4}{x} \\ & \Rightarrow 4 \sqrt{3}=x \end{aligned} \] In right \(\displaystyle \Delta \mathrm{PBQ}\), \[\begin{aligned} & \tan 60^{\circ}=\sqrt{3}=\frac{4}{y} \\ & \Rightarrow y=\frac{4}{\sqrt{3}} \end{aligned} \] Width of river \(\displaystyle =\mathrm{x}+\mathrm{y}=4 \sqrt{3}+\frac{4}{\sqrt{3}}=\frac{16}{3} \sqrt{3} \mathrm{~m}\)