Question

A stationary balloon is observed from three points $A,B$ and $C$ on the plane ground and is found that its angle of elevation from each point is $60^{\circ }.$ If $\angle ABC=30^{\circ }$ and $AC=5$ meters, then the height of the balloon from the ground is

• A. $5\sqrt{3}$ meters
• B. $\frac{5}{4}\sqrt{3}$ meters
• C. $\frac{5}{\sqrt{3}}$ meters
• D. $\frac{5}{4\sqrt{3}}$ meters

Answer 1

Answer: (A)

Let, $PQ$ be vertical with height $h$ meters and at its end $Q$ there is a balloon and $P$ is on the ground
$\Rightarrow PA=PB=PC=hcot60^{\circ }=\frac{h}{\sqrt{3}}$
$\therefore$ $P$ is the circumcentre of $\Delta ABC$ and its circumradius $\left(R\right)=\frac{h}{\sqrt{3}}$

In $\Delta ABC$ ,
$2R=\frac{5}{sin30^{\circ }}\Rightarrow 2\frac{h}{\sqrt{3}}=\frac{5}{sin30^{\circ }}$
$\therefore h=5\sqrt{3}$ meters.