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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