From a point at a height h m above a lake- the angle of elevation of a cloud is - and the angle of depression of its reflection in the lake is - The height of the cloud above the surface of the lake is

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Calculating Heights and Distances

From a point ...

Question

From a point at a height $h$ m above a lake, the angle of elevation of a cloud is $α$ and the angle of depression of its reflection in the lake is $β$ . The height of the cloud above the surface of the lake is

\frac{h cos (α + β)}{cos (α - β)} m

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\frac{h sin (α + β)}{cos (α - β)} m

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\frac{h sin (α - β)}{cos (α + β)} m

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\frac{h sin (α + β)}{sin (β - α)} m

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Solution

The correct option is D $\frac{h sin (α + β)}{sin (β - α)} m$

Let $H$ be the height of the cloud from the point above $h m$ from the lake surface.
Now the distance of the reflection from the lake surface be $H + h m$
Now, In $△ C D E,$
$\Rightarrow tan α = \frac{D E}{C E}$
$\Rightarrow C E = \frac{D E}{tan α} = \frac{H}{tan α} ⟶ (1)$
In $△ C E D^{'},$
$\Rightarrow tan β = \frac{E D^{'}}{C E}$
$\Rightarrow C E = \frac{E D^{'}}{tan β} = \frac{H + 2 h}{tan β} ⟶ (2)$
From equation $(1) & (2),$ we get
$\Rightarrow \frac{H}{tan α} = \frac{H + 2 h}{tan β}$
$\Rightarrow H tan β = H tan α + 2 h tan α$
$\Rightarrow H (tan β - tan α) = 2 h tan α$
$\Rightarrow H = \frac{2 h tan α}{tan β - tan α}$
Now, height of the cloud above the lake surface
$\Rightarrow H + h = \frac{2 h tan α}{tan β - tan α} + h$
$= \frac{2 h tan α + h tan β - h tan α}{tan β - tan α}$
$= \frac{h (tan α + tan β)}{tan β - tan α}$
$= h ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{\frac{sin α}{cos α} + \frac{sin β}{cos β}}{\frac{sin β}{cos β} - \frac{sin α}{cos α}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
$= h ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{\frac{sin α cos β + cos α sin β}{cos α cos β}}{\frac{sin β cos α - cos β sin α}{cos α cos β}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
Hence, the answer is $h ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{\frac{sin α cos β + cos α sin β}{cos α cos β}}{\frac{sin β cos α - cos β sin α}{cos α cos β}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$ $= \frac{h sin (α + β)}{sin (β - α)} m$

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From a point at a height h m above a lake, the angle of elevation of a cloud is α and the angle of depression of its reflection in the lake is β. The height of the cloud above the surface of the lake is

From a point at a height $h$ m above a lake, the angle of elevation of a cloud is $α$ and the angle of depression of its reflection in the lake is $β$ . The height of the cloud above the surface of the lake is