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Question
Of the two trees are on either side of a river, one of them is 50m high. From the top of this tree the angles of depression of the top and the foot of the other tree are 30° and 60° respectively. Find the width of the river and the height of the other tree.
Solution
Let AB and CD be the two trees.
In ΔAEC,
`tan30^circ = "EA"/"EC"`
⇒ `1/sqrt(3) = "EA"/"EC"`
⇒ `"EC" = sqrt(3)"EA"` ....(1)
In ΔABD,
`tan60^circ = "AB"/"BD" = sqrt(3)`
⇒ `50/"BD" = sqrt(3)`
⇒ `"BD" = 50/sqrt(3)`
Thus , the width of the river is `"BD" = 50/sqrt(3) = 28.8` m.
From (1),
`"EA" = "EC"/sqrt(3) = "BD"/sqrt(3) = 50/3 = 16.67`
Height of the other tree =CD = 50 - EA= 50 - 16.67 = 33.33 m
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