Question

The height of a light house is $85m$. An observer on it found the angle of depression to a boat to be ${30}^o$, which was sailing towards the light house. After $2$ minutes, the angle of depression for the same boat was ${60}^o$. What was the speed of the boat in $km/hr$?

Answer 1

Let $AB$ be the lighthouse, $C$ be the initial point and $D$ be the final point of the boat.

It can be observed that $△ABC$ and $△ABD$ are right angled triangles.

In $△ABC$,

$\tan {60}^0=\frac{85}{y}$

$\sqrt{3}=\frac{85}{y}$

$y=\frac{85}{\sqrt{3}}m$

In $△ABD$

$\tan {30}^0=\frac{85}{x+y}$

$\frac{1}{\sqrt{3}}=\frac{85}{x+y}$

$x+y=85\sqrt{3}$

$x=85\sqrt{3}-\frac{85}{\sqrt{3}}$

$=\left(\frac{255-85}{\sqrt{3}}\right)$

$x=\frac{170}{\sqrt{3}}m$

Distance travelled in $2$ minutes $x=\frac{170}{\sqrt{3}}m$

distance travelled in $1$ minutes $x=\frac{170}{2\sqrt{3}}m/min$

$=\frac{170\times 60}{1000\sqrt{3}\times 2}km/hr$

$=1.24km/hr$

Hence,Speed of boat is $1.24km/hr$