Question

A man on a cliff observes a boat at an angle of depression 30^o, which is sailing towards the point of the shore immediately beneath him. Three minutes later the angle of depression of the boat is found to be 60^o . Assuming that the boat sails at a uniform speed, determine how much more time it will take to reach the shore.

Answer 1

Let the speed of the boat be v units and the time taken by the boat to reach the shore be t.
Also,
Let:
AB = h units
Now,
BC = tv and CD = 3v
And,
BD = BC + CD = tv + 3v

$$\begin{gathered} \mathrm{In}△ABD, \\ \tan 30°=\frac{AB}{BD}=\frac{h}{3v+tv} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{tv+3v} \\ \Rightarrow \sqrt{3}h=tv+3v...\left(\mathrm{i}\right) \\ \mathrm{And}, \\ \mathrm{In}△ABC, \\ \tan 60°=\frac{AB}{BC} \\ \Rightarrow \sqrt{3}=\frac{h}{tv} \\ \Rightarrow h=\sqrt{3}vt...\left(\mathrm{ii}\right) \\ \mathrm{From}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get}: \\ \sqrt{3}\times \sqrt{3}vt=tv+3v \\ \Rightarrow 3vt=vt+3v \\ \Rightarrow 2vt=3v \\ \Rightarrow t=\frac{3}{2}\mathrm{mins}=1\min 30\mathrm{s} \end{gathered}$$