Question

From an airplane flying vertically above a horizontal road, the angle of depression of two consecutive kilometer stones on the same side of the airplane are observed to be ${30}^{\circ }$ and ${60}^{\circ }$, respectively. The height at which the airplane is flying (in km) is:

• A. $\frac{4}{\sqrt{3}}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{2}{\sqrt{3}}$
• D. $2$

Answer 1

Answer: (B)

$\frac{\sqrt{3}}{2}$

Let the distance of two consecutive stones are $x$, $x+1$.
In $\Delta BCD$, we have
$\tan {60}^o=\frac{h}{x}$
$\Rightarrow x=\frac{h}{\sqrt{3}}$ .....(i)
In $\Delta ABC$, we have
$\tan {30}^o=\frac{h}{x+1}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x+1}$
$\Rightarrow \frac{h}{\sqrt{3}}+1=\sqrt{3}h$ ......[from equation (i)]
$\Rightarrow \frac{2h}{\sqrt{3}}=1$
$\Rightarrow h=\frac{\sqrt{3}}{2}km$