Question

A boat is rowed away from a cliff $150$ m high At the top of the the cliff the angle of depression of the boat change from ${60}^0$ to ${45}^0$ in $2.5$ minutes The speed of the boat (in m/sec) is

• A. $1+\frac{1}{\sqrt{3}}$
• B. $1-\frac{1}{\sqrt{3}}$
• C. $\sqrt{3}+3$
• D. $2\sqrt{3}-3$

Answer 1

The correct option is B $1-\frac{1}{\sqrt{3}}$

Let AB be cliff C and D be two position of the boat
$AB=150m,\angle ACB={60}^{0}and\angle ADB={45}^0$

$AC=AB\cot {60}^0$

$\Rightarrow AC=\frac{150}{\sqrt{3}}m$

$AD=AB\cot {45}^0$

$\Rightarrow AD=150m$

$\therefore CD=150-\frac{150}{\sqrt{3}}$

$=\frac{150}{\sqrt{3}}(\sqrt{3}-1)$ which is covered in 150sec.

$\therefore$ Speed of the boat in m/sec

$=\frac{150\left(\sqrt{3-1}\right)}{\sqrt{3}\times 150}$

$=\frac{\sqrt{3}-1}{\sqrt{3}}$

$=1-\frac{1}{\sqrt{3}}$