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The angle of elevation of a cloud from a point $h$ meters above the surface of a lake is $30 °$ and the angle of depression of its reflection is $60 °$ . Then the height of the cloud above the surface of the lake is

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h \sqrt{3}

h

\sqrt{2} h

2 h

answer is D.

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Detailed Solution

It is given that the angle of elevation of the cloud is

30 °

from a point which is

h

meters above the surface of a lake and the angle of depression of its reflection on the lake is

60 °

.
According to the given data the figure has been drawn below.
Question Image

In the figure, C is the position of the cloud and A is the point of observation.
So

∠ CAB = 30 °

and

∠ BAE = 60 ° .

Now

CD = DE = x

(let)
In

ΔABC,

tan θ = P e r p e n d i c u l a r b a s e

\frac{CB}{AB} = \tan 30 °

From the figure it is clear that CB = CD - BD.
Now CE = x and BD = AF = h (as they are opposite sides of the rectangle ABDF)
So, CE = x-h.
Hence we get,

\Rightarrow (x - h) = \frac{AB}{\sqrt{3}} . . . (1)

∵ tan 30 ° = 13

Similarly from

ΔABE,

we get,

tan θ = P e r p e n d i c u l a r b a s e

\frac{BE}{AB} = \tan 60 °

(x + h) = AB \sqrt{3}

……(2)

∵ tan 60 ° = 3

Dividing (1) by (2) we get,

\frac{x - h}{x + h} = \frac{AB}{\sqrt{3} . \sqrt{3} . AB}

\Rightarrow \frac{x - h}{x + h} = \frac{1}{3}

\Rightarrow 3 x - 3 h = x + h

\Rightarrow 2 x = 4 h

\Rightarrow x = 2 h

∴

Height of cloud

= 2 h .

So the correct option is 4.

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