The angle of elevation of cloud from a point 60 m above a lake is ` 30^(@)` and the angle of depression of the reflection of cloud in the lake is ` 60^(@)` . Find the height of the cloud .

Answer

The height of the cloud is 120 m.

Explanation

Let AB be the surface of the lake and H be the point of observation Let C be the position of the cloud above the lake and D be its reflection in the lake .
` :. BC = BD " " ` …(1)
AH = MB = 60 m . Let CM be x m .
` CB = CM +MB " " ...(C-M-B)`
` x + 60`
` :. CB = (x+60) m " "`...(2)
From (1) and (2) ,
`BD = (x+60)m " "` ...(3)
` MD = MB +BD " " ... (M-B-D)`
` = (60 + x + 60 )m " "` ...[ From (3)]
` :. MD = (x + 120 )m" "` (4)
In right angled ` Delta CHM,`
`tan angle CHM = tan 30^(@) = (CM)/(HM)`
` :. HM = x sqrt(3))`
In right angled `Delta DHm, tan angle DHM = tan 60^(@) = (MD)/(HM)`
` :. sqrt(3) = (x+120)/(HM)" "` ...[From (4)]
` :. HM = (x+120)/(sqrt(3)) " " ` ...[ From (4)]
` :. HM = (x+120)/(sqrt(3))" "` ...(6)
From (5) and (6) ,
` x sqrt(3) + (x+120)/(sqrt(3))`
` :. 3x = x+20 `
` :. 2x = 120 " " :. x = 60`
The height of the cloud `(CB) = (x+60) m" "` ... [ From (2)]
` = (60 + 60 ) m = 120 m`