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Question

A man rowing a boat away from a lighthouse which is 100m high takes 2 mins to change the angle of elevation of the lighthouse from 60 to 45. Calculate the speed of the boat.

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Solution

Let AB be the lighthouse and C, D be the positions of the man when the angle of elevation changes from 60 to 45.
The man has covered a distance CD in 2 min. As per question the height of the lighthouse is 100m. So, AB = 100m.

Speed=DistanceTime

In 2 mins, the man moves from point C to D.
Speed=CD2m/minIn ΔABC, we havetan 60=ABBC 3=100BCBC=1003×33
BC=10033 .... (i)

In ΔABD, we have
tan 45=ABBD1=100BDBD=100Also CD=DBBC=10010033 [using (i)]=100(133)=100(333) Speed=CD2=100(333)2=50(333)m/min

Hence, the required speed of boat is 503(33) m/min


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