Question

A moving boat is observed from the top of a 150 m high cliff moving away from cliff. The angle of depression of the boat changes from 60 degrees to 45 degrees in 2 minutes. Find the speed of the boat in m/h.

Answer 1

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Dear Student,

Please find below the solution to the asked query:

We form our diagram from given information , As :

Here , Height of cliff =  AC = 150 m 

And boat's initial position is at ' D '  and position of boat after 2 minutes is at ' B ' .

We know : tan θ = OppositeAdjacent , So

In triangle ABC we get :

tan (∠ ABC) = ACAB⇒tan 45° = 150AB⇒1 = 150AB     ( We know : tan 45° =1 )⇒AB = 150 


And in triangle ADC we get :

tan (∠ ADC) = ACAD⇒tan 60° = 150AD⇒3‾√ = 150AD     ( We know : tan 60° =3‾√ )⇒AD = 1503√⇒AD = 1503√×3√3√⇒AD = 1503√ 3⇒AD = 503‾√  = 50 ×1.732 = 86.6
So,

Distance travelled by boat in 2 minutes = AB - AD =  150 - 86.6 = 63.4 m

And time taken to cover distance 63.4 m = 2 minutes = (2 ×160)hr = (130)hr

We know :  Speed = DistanceTime

Therefore,

Speed of boat = 63.4130 = 63.4 × 30 = 1902 m/hr              ( Ans )


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