Question

A boat is being rowed away from a cliff 150 m high. At the top of the cliff the angle of depression of the boat changes from ${60}^o$ to ${45}^o$ in 2 minutes. The speed of the boat is

• A. 2 km/hr
• B. 1.9 km/hr
• C. 2. km/hr
• D. 3 km/hr

Answer 1

The correct option is B 1.9 km/hr

Height of the lighthouse $AB=150m$

Now, $\angle ABC={90}^o$ and $\angle ACB={45}^o$ then, $\angle BAC={45}^o$

Now, in right angled $△ACB,$

$\tan {45}^o=\frac{AB}{CB}$

$1=\frac{150}{CB}$

$\Rightarrow$ $CB=150m$ ------ ( 1 )

In right angled $△ADB,$

$\tan {60}^o=\frac{AB}{BD}$

$\sqrt{3}=\frac{150}{BD}$

$\Rightarrow$ $BD=\frac{150}{\sqrt{3}}=\frac{150\sqrt{3}}{3}=50\sqrt{3}m$ ----- ( 2 )

Distance between $CD=CB-BD$

$=(150-50\sqrt{3})m$

$=150-86.6m$

$=63.4m$

Time taken to cover $63.4m=2min$

$\therefore$ Speed of the boat $=\frac{63.4}{2\times 60}=0.53m/sec$

$=0.53\times \frac{18}{5}$

$=1.9km/hr$