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. 

Answer 1

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