Question

The angle of elevation of a cloud from a point 60m above a lake is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud. 

Answer 1

Let C be the cloud and D be its reflection . Let the height of the cloud is h metres .

BC = BD = h

BQ = AP = 60m 

∴ CQ = h - 60 and DQ = h + 60

In ΔCQP,

`"PQ"/"CQ" = cot30^circ`

⇒ `"PQ"/("h" - 60) = sqrt(3)`

⇒ `"PQ" = sqrt(3)("h" - 60)"`   ....(i)

In ΔDQP,

`"PQ"/"DQ" = cot60^circ`

⇒ `"PQ"/("h" + 60) = 1/sqrt(3)`

⇒ `"PQ" = 1/sqrt(3)("h" + 60)`   ..(ii)

From (i) and (ii),

⇒ `sqrt(3)("h" - 60) = 1/sqrt(3)("h" + 60)`

⇒ 3h - 180 = h + 60

⇒ 2h = 240

⇒ h = 120

Thus , the height of the cloud is 120 m.