Problem Statement
The angle of elevation of a cloud from a point $h$ meters above the surface of 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 surface of the lake is:
$h \left( \frac{\tan \beta + \tan \alpha}{\tan \beta - \tan \alpha} \right)$
Verified Solution & Marking Scheme
Set up Geometric Diagram Relations
Let the surface of the lake be the horizontal reference line.
- Point of observation $O$ is at height $h$ above the lake surface.
- Let the cloud $C$ be at height $H$ above the lake surface.
- Height of cloud above observer $O$ is $H - h$.
- The reflection $R$ of the cloud in the lake is at depth $H$ below the surface.
- Distance of reflection $R$ below the observer $O$ is $H + h$.
- Let the horizontal distance from the observer to the vertical line of the cloud be $d$.
Write Trigonometric Ratios for Elevation and Depression
From the right triangle formed with the cloud:
$\tan \alpha = \frac{H - h}{d} \implies d = \frac{H - h}{\tan \alpha}$ ...(1)
From the right triangle formed with the reflection:
$\tan \beta = \frac{H + h}{d} \implies d = \frac{H + h}{\tan \beta}$ ...(2)
Equate Horizontal Distance d and Solve for H
Equating (1) and (2):
$\frac{H - h}{\tan \alpha} = \frac{H + h}{\tan \beta}$
$(H - h) \tan \beta = (H + h) \tan \alpha$
$H \tan \beta - h \tan \beta = H \tan \alpha + h \tan \alpha$
$H (\tan \beta - \tan \alpha) = h (\tan \beta + \tan \alpha)$
$H = h \left( \frac{\tan \beta + \tan \alpha}{\tan \beta - \tan \alpha} \right)$.