Question

A man on the top of a rock observed a boat coming towards the rock with a uniform speed. It takes $15$ minutes for the angle of depression to change from ${60}^{\circ }$ to ${30}^{\circ }$ then what time will the boat take to reach the shore?

• A. $5$ mintues
• B. $10$ mintues
• C. $7\frac{1}{2}$ mintues
• D. $2\frac{1}{2}$ mintues

Answer 1

The correct option is C $7\frac{1}{2}$ mintues

Let, the Height of the Rock be $h$

so, $tan{30}^o=\frac{h}{BD}\Rightarrow BD=h\sqrt{3}tan{60}^o=\frac{h}{BC}\Rightarrow BC=\frac{h}{\sqrt{3}}$

Now, Distance travelled by Boat in 15 minutes $=h\sqrt{3}-\frac{h}{\sqrt{3}}$

$\Rightarrow v=\frac{h\sqrt{3}-\frac{h}{\sqrt{3}}}{15}\Rightarrow v=\frac{2h}{15\sqrt{3}}$

Now to reach the Shore it has to travel Distance $\frac{h}{\sqrt{3}}$ in Time $t$

$\Rightarrow t=\frac{\frac{h}{\sqrt{3}}}{\frac{2h}{15\sqrt{3}}}=7\frac{1}{2}minutes$