Question

The angle of elevation of a cloud from a point 120m above a lake is 30° and and the angle of depression of it's reflection in the lake is 60° . Find the length of the cloud

Answer 1

$\boxed{height \: of \: cloud \: from \: lake = 240m}$

EXPLANATION :-

According to attachment provided with the answer :-

$in \triangle \: BCE : - \\ \\ \tan60\degree = \frac{EC}{BC} = \frac{240+x}{BC} \\ \\ \\ BC = \frac{240+x}{ \sqrt{3} }$

Now,

$in \triangle \: ACB : - \\ \\ \tan(30) = \frac{AC}{BC} = \frac{x\sqrt{3}}{240+x} \\ \\ \\ \frac{x\sqrt{3}}{240+x}=\frac{1}{\sqrt{3}}\\ \\ \\ 3x=240+x \\ \\ \\ 2x =240 \\ \\ \\ x = \frac{240}{2}=120$

Thus,

Height of Cloud from lake is :-

120 + x :-

$\bold{120 + 120 = 240m}$

Answer 2

Solution:-

Let ED be the Surface of the Lake.

EB= CD = 120m.

BC = x m.

& AC = h m.

Now,

AC + CD = DF ( The Height of Cloud is equal as its Reflection in the lake.)

=> DF = ( 120 + h) m.

Now,

In rt. ∆ ABC,

Tan 30° = h / x

=> 1/ √3 = h/ X

=> X = h √3 m.____________(1)

In rt. ∆ BCF,

Tan 60° = CF / BC

=> √3 = ( 120 + h + 120) / X

=> X = (240 + h)/ √3________(2).

Equating eq (1) & (2). we get,

√3 = ( 240 + h) / √3

=> 3h = 240 + h

=> 2h = 240 m.

=> h = 120m.

Now,

Height of the Cloud = h + 120 = 120 + 120 = 240m.

Hence,

Height of Cloud = 240m.