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Question
The angle of depression of a boat moving towards a diff is 30°. Three minutes later the angle of depression of the boat is 60°. Assuming that the boat is sailing at a uniform speed, determine the time it will take to reach the shore. Also, find the speed of the boat in m/second if the cliff is 450m high.
Solution
Let AB be the diff. Then , AB = 450 m
Initial position of boat is C , which changes to D after 3 minutes.
In ΔADB
`"AB"/"DB" = tan60^circ`
`450/"DB" = sqrt(3)`
`"DB" = 450/sqrt(3)`
In ΔABC
`"AB"/"BC" = tan30^circ`
`450/("BD + DC") = 1/sqrt(3)`
`450sqrt(3) = "BD + DC"`
`450sqrt(3) = 450/sqrt(3) + "DC"`
`"DC" = 450sqrt(3) - 450/(sqrt(3)) = 450(sqrt(3) - 1/sqrt(3))`
= `900/sqrt(3) = 900/sqrt(3) xx sqrt(3)/sqrt(3) = 300sqrt(3)`
Time taken by car to travel DC distance (`"i.e.,"300sqrt(3)`) = 3 minutes
Time taken by car to travel DB distance `("i.e". 450/sqrt(3))`
= `3/(300sqrt(3)) xx 450/sqrt(3) = 450/300 = 1.5`
Thus , the time it will take to reach the shore is 1 min 30 secs.
Speed of the boat = `"Distance"/"Time"`
= `(300sqrt(3))/3 = 100sqrt(3) = 100 xx 1.732 = 173.2` m/min
= `173.2/60` m/sec = 2.9 m/sec
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