Question

From an aeroplane above a straight road the angles of depression of two positions at a distance 20 m apart on the road are observed to be ${30}^o$ and ${45}^o$. The height of the aeroplane above the ground is :

• A. $10\sqrt{3}m$
• B. $10(\sqrt{3}-1)m$
• C. $10(\sqrt{3}+1)m$
• D. $20m$

Answer 1

The correct option is B $10(\sqrt{3}-1)m$

From the figure

$\frac{h}{20-x}=\tan {30}^0$

$\frac{h}{20-x}=\frac{1}{\sqrt{3}}$

$\sqrt{3}h=20-x$ ....... $\left(1\right)$

and $\frac{h}{x}=\tan {45}^0$

$h=x$ ....... $\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$

$\sqrt{3}h=20-h\Rightarrow \frac{20}{\sqrt{3}+1}=h$

$\Rightarrow h=10(\sqrt{3}-1)$