Question

A tower of height $60m$ stands on the top of a cliff. An observer in a boat finds that the elevation of the top of the tower is ${60}^{\circ }$ and that of the top of the cliff is ${30}^{\circ }$. Find the height of the cliff, and the distance of the boat from the foot of the cliff.

Answer 1

If $h$ is the height of cliff $BC$ on which tower $AB$ of height $60m$ stands and $D$ is the point at a distance $x$ from the base of the cliff.

In $△BCD,$

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

And, in $△BCD,$

$\begin{array}{cc}\tan {60}^{\circ }& =\frac{AC}{CD}\\ \sqrt{3}& =\frac{60+h}{x}\\ \sqrt{3}x& =60+h\\ \sqrt{3}\times \sqrt{3}h& =60+h\dots \left[\text{By using}\right(i\left)\right]\\ 3h-h& =60\\ 2h& =60\\ h& =30m\end{array}$

By using $\left(i\right),$

$x=\sqrt{3}\times 30m$

$=30\sqrt{3}m$