Question

A man on a cliff observes a boat at an angle of depression of ${30}^o$ 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 ${60}^o$. Find the total time taken by the boat to reach the shore.

Answer 1

Let the speed be $x$ $km/h$

Let the time be $t$

In $△ADC,$

$\tan 30°=\frac{xt}{h}$

$\frac{1}{\sqrt{3}}=\frac{xt}{h}$

$h=\sqrt{3}xt$

In $△ABD,$

$\tan 60°=\frac{x(6+t)}{h}$

$h=\frac{x(t+6)}{\sqrt{3}}$

$\sqrt{3}xt=\frac{x(t+6)}{\sqrt{3}}$

$3t=6+t$

$t=3$ $seconds$

$\therefore$ total time taken$=6+3=9$ $seconds$