Question

A man on a cliff observes a boat at an angle of depression ${30}^o$ which is sailing towards the shore to the point immediately beneath him. $3$ minutes later the angle of depression of the boat is found to be ${60}^o$. Assuming that the boat sails at a uniform speed, the time taken by the boat to reach the shore is:

• A. $1.5$ min
• B. $4.5$ min
• C. $6$ min
• D. $6.5$ min

Answer 1

The correct option is B $4.5$ min

Let $AB$ be the cliff, speed be $s$, total time taken be $T$, total distance be $X$

The distance covered in $3$ min is $x=3s$

Height of cliff is $h$

From $\Delta ABD$

$⟹\tan {30}^o=\frac{AB}{BD}$

$⟹\frac{1}{\sqrt{3}}=\frac{AB}{X}$

$⟹AB=\frac{X}{\sqrt{3}}\cdots \left(1\right)$

From $\Delta ABC$

$⟹\tan {60}^o=\frac{AB}{BC}$

$⟹\sqrt{3}=\frac{AB}{X-x}$

$⟹AB=X-x\times \sqrt{3}\cdots \left(2\right)$

From $\left(1\right),\left(2\right)$

$⟹\frac{X}{\sqrt{3}}=(X-3s)\times \sqrt{3}$

$⟹X=3X-9s$

$⟹2X=9s$

$⟹2\left(sT\right)=9s$

$⟹T=\frac{9}{2}=4.5$ min