Problem Statement
The angle of elevation of a cloud from a point $h$ meters above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. Prove that the height of the cloud above the lake is:
$\frac{h(\tan\beta + \tan\alpha)}{\tan\beta - \tan\alpha}$
Verified Solution & Marking Scheme
Set Up Heights Model
Let the lake surface be $y = 0$. Observer is at point $A$ at height $h$ above water.
Let the height of the cloud $C$ above water be $H$.
Height of cloud above observer $= H - h$.
Since the surface of the lake acts as a plane mirror, the reflection of the cloud $C'$ is at distance $H$ below the water surface.
Depth of reflection below observer $= H + h$.
Relate Horizontal Distance d
Let the horizontal distance from observer to the vertical line of the cloud be $d$:
$\tan\alpha = \frac{H - h}{d} \implies d = \frac{H - h}{\tan\alpha}$
$\tan\beta = \frac{H + h}{d} \implies d = \frac{H + h}{\tan\beta}$
Equate d and Solve for H
$\frac{H - h}{\tan\alpha} = \frac{H + h}{\tan\beta} \implies (H - h)\tan\beta = (H + h)\tan\alpha$
$H\tan\beta - h\tan\beta = H\tan\alpha + h\tan\alpha$
$H(\tan\beta - \tan\alpha) = h(\tan\beta + \tan\alpha)$
$H = \frac{h(\tan\beta + \tan\alpha)}{\tan\beta - \tan\alpha}$