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

Answer 1

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`