Question : A person from the top of a hill observes a vehicle moving towards him at a uniform speed. It takes 10 minutes for the angle of depression to change from $45^{\circ}$ to $60^{\circ}$. After this, the time required by the vehicle to reach the bottom of the hill is:
Option 1: 12 minutes 20 seconds
Option 2: 13 minutes
Option 3: 13 minutes 40 seconds
Option 4: 14 minutes 24 seconds
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Correct Answer: 13 minutes 40 seconds
Solution :
Let, $CD$ = 10 units
In $\Delta ABC$,
$\tan 45^{\circ} = \frac{AB}{BC}$
$\Rightarrow BC = AB \quad................(1)$
In $\Delta ABD$,
$\tan 60^{\circ} = \frac{AB}{BD}$
$\Rightarrow \sqrt{3} = \frac{AB}{BD}$
$\Rightarrow AB = \sqrt{3} BD$
$\Rightarrow BC = \sqrt{3} BD \quad [\text{using (1)}]$
$\Rightarrow BC = BD + CD$
$\Rightarrow \sqrt{3} BD - BD = CD$
$\Rightarrow BD(\sqrt{3} - 1) = 10$
$\Rightarrow BD = \frac{10}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1}$
$\Rightarrow BD = \frac{10(\sqrt{3} + 1)}{2}$
$\Rightarrow BD = 5(1.732 + 1)$
$\Rightarrow BD = 5 \times 2.732$
$\Rightarrow BD = 13.66$ units
$\therefore $ Time required to travel 10 units = 10 minutes
$\Rightarrow$ Time required to travel 13.66 units = 13.66 minutes = 13 minutes 40 seconds approx.
Hence, the correct answer is 13 minutes 40 seconds.
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