Question

The angle of depression of two post $P$ and $Q$ at a distance of $2$ meter on the same side of a road from a balloon $\left(B\right)$ vertically over the road are observed to be ${45}^{\circ }$ and ${60}^{\circ }$. What is the height of the balloon?

• A. $3-\sqrt{3}$
• B. $\sqrt{3}-1$
• C. $3+\sqrt{3}$
• D. $\sqrt{3}+1$

Answer 1

Answer: (C)

$3+\sqrt{3}$

let the height of the balloon be $h$ and the point on ground be S

In $△PBS$

$\tan 45=\frac{BS}{PS}$

$BS=PS=h$

In $△QBS$

$\tan 60=\frac{BS}{QS}$

$BS=QS\sqrt{3}$

$QS=\frac{h}{\sqrt{3}}$

Now, $PS-QS=2$

$h-\frac{h}{\sqrt{3}}=2$

$h=\frac{2\sqrt{3}}{\sqrt{3}-1}$

$h=\frac{2\sqrt{3}(\sqrt{3}+1)}{3-1}$ (Rationalising)

$h=3+\sqrt{3}$