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Question
If the angle of elevation of a cloud from a point 200 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°, then the height of the cloud above the lake is
Options
200 m
500 m
30 m
400 m
Solution
Let AB be the surface of the lake and P be the point of observation. So AP=60 m.
The given situation can be represented as,
Here,C is the position of the cloud and C' is the reflection in the lake. Then `CB=C'B`.
Let `PM` be the perpendicular from P on CB. Then `∠CPM=30°` and`∠C' PM=60°` and .
Let`CM=h` `PM=x`, , then`CB=h+200` and`C'B=h+200`
Here, we have to find the height of cloud.
So we use trigonometric ratios.
In ,`ΔCMP`
`⇒ tan 30°=CM/PM`
`⇒1/sqrt3=h/x`
`⇒x=sqrt3h`
Again in `ΔPMC`
`⇒ tan 60°= (C'M)/(PM)`
`⇒sqrt3=(C'B+BM)/(PM)`
`⇒sqrt3=(h+200+200)/x`
`⇒ sqrt3x=h+400`
Put `x=sqrt3h`
`⇒3h=h+400`
`⇒ 2h=400`
`⇒h=200`
Now,
`⇒ CB=h+200`
`⇒ CB=200+200`
`⇒ CB=400`
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