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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.
Solution
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.
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