Question : A man on the top of a tower, standing on the seashore, finds that a boat coming towards him takes 10 minutes for the angle of depression to change from 30° to 60°. How soon does the boat reach the seashore?
Option 1: 5 minutes
Option 2: 7 minutes
Option 3: 10 minutes
Option 4: 15 minutes
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Correct Answer: 5 minutes
Solution :
Given: The boat takes 10 minutes for the angle of depression to change from 30° to 60°.
We know the trigonometric ratio, $\tan\theta=\frac{\text{Height}}{\text{Base}}$.
In $\triangle ABC$, $\tan \ 30° =\frac{AB}{BC}$
$⇒\frac{1}{\sqrt3} = \frac{AB}{BC}$
$⇒AB=\frac{BC}{\sqrt3}$-------(equation 1)
In $\triangle ABD$, $\tan \ 60° =\frac{AB}{BD}$
$⇒\sqrt3 = \frac{AB}{BD}$
$⇒AB = \sqrt3 BD$------(equation 2)
From equation 1 and equation 2, we get,
$⇒\frac{BC}{\sqrt3}=\sqrt3 BD$
$⇒BC= 3 BD$
Also, $BC= BD+CD$
$⇒3BD=BD+10x$ [Here $CD= 10x$, if the speed of boat is $x$ m/minute]
$⇒2BD=10x$
$⇒BD=5x$
So, Time taken = $\frac{5x}{x}$ = 5 minutes
Hence, the correct answer is 5 minutes.
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