A man on cliff observes a boat at an angle of depression of 30 degree which is approaching the shore - Maths - Some Applications of Trigonometry

Question

A man on cliff observes a boat at an angle of depression of 30
degree which is approaching the shore to the point immediately
beneath the observer with a uniform speed Six minutes later , the
angle of depression of the boat is found to be 60 degree Find the
total time taken by the boat to reach the shore

Priyanka Kedia · Accepted Answer

Let the uniform speed be x and the distance to be travelled by the boat to reach the soar is y

$I n Δ A B C, {\tan 30}^{0} = \frac{A B}{B C} \Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{y + 6 x} \Rightarrow h = \frac{y + 6 x}{\sqrt{3}} ...... (1)$

And,

$I n Δ A B D, {\tan 60}^{0} = \frac{A B}{B B} \Rightarrow \sqrt{3} = \frac{h}{y} \Rightarrow h = y \sqrt{3} ...... (2)$

equating equation (1) and (2), we get,

$\frac{y + 6 x}{\sqrt{3}} = y \sqrt{3} \Rightarrow y + 6 x = 3 y \Rightarrow 6 x = 2 y \Rightarrow y = 3 x$

Hence, the time taken to reach the soar is 3 minute