A man on a cliff observes a boat at an angle of depression of $\displaystyle 30$° which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is found to be $\displaystyle 60$°. Find the time taken by the boat from here to reach the shore.

\[\begin{aligned} & \text { In } \triangle Q A B \\ & \tan 60^{0}=\sqrt{3}=\frac{h}{x} \\ & \mathrm{~h}=\sqrt{3} \mathrm{x} \ldots \ldots \ldots \ldots \ldots \end{aligned} \] \[\text { In } \triangle P A B \] \[\begin{aligned} & \tan 30^{0}=\frac{1}{\sqrt{3}}=\frac{h}{y+x} \\ & \mathrm{y}+\mathrm{x}=\sqrt{3} \mathrm{~h} \ldots \ldots \ldots . \end{aligned} \] solving (i) and (ii) \[y=2 x \] Time taken by the boat from Q to \(\displaystyle \mathrm{A}=\frac{1}{2} \times 6=3 \mathrm{~min}\).