Question

The angle of elevation of a cloud from a point $60$ $m$ above the surface of the water of a late is ${30}^o$ and the angle of depression of its shadow from the same point in water of lake is ${60}^o$. Find the height of the cloud from the surface of water.

Answer 1

In the above figure, $C$ is the cloud and $F$ is its reflection on the surface of the lake $AB.$ And $E$ is the point of observation.

In $△CDE,$

$\begin{array}{cc}\tan {30}^{\circ }& =\frac{CD}{DE}\\ \frac{1}{\sqrt{3}}& =\frac{x}{y}\\ y& =\sqrt{3}x\dots \left(i\right)\end{array}$

According to laws of reflection, $BC$ will be equal to $BF.$

So,

$BF=BC$

$=BD+DC$

$=60+x$

In $△EDF,$

$FD=BF+BD$

$=60+x+AE$

$=60+x+60$

$=120+x$

So,

$\begin{array}{cc}\tan {60}^{\circ }& =\frac{FD}{DE}\\ \sqrt{3}& =\frac{120+x}{y}\\ \sqrt{3}\times \sqrt{3}x& =120+x\dots \left[\text{By using}\right(i\left)\right]\\ 3x-x& =120\\ 2x& =120\\ x& =60m\end{array}$

So, the height of the cloud from the surface of the lake will be,

$BC=60+x$

$=60+60$

$=120m$