Question : If the angle of elevation of a balloon from two consecutive kilometre stones along a road are 30° and 60° respectively, then the height of the balloon above the ground will be:
Option 1: $\frac{\sqrt{3}}{2}$ km
Option 2: $\frac{1}{2}$ km
Option 3: $\frac{2}{\sqrt{3}}$ km
Option 4: $3\sqrt{3}$ km
Correct Answer: $\frac{\sqrt{3}}{2}$ km
Solution :
Let the height of the balloon be $h$.
$\tan60°=\frac{h}{x}$
⇒ $\sqrt{3}=\frac{h}{x}$
⇒ $x=\frac{h}{\sqrt{3}}$ ------(1)
$\tan30°=\frac{h}{x+1}$
⇒ $\frac{1}{\sqrt{3}}=\frac{h}{\frac{h}{\sqrt{3}}+1}$
⇒ $\frac{1}{\sqrt{3}}=\frac{\sqrt{3}h}{h+\sqrt{3}}$
⇒ $3h=h+\sqrt{3}$
⇒ $2h=\sqrt{3}$
⇒ $h=\frac{\sqrt{3}}{2}$ km
Hence, the correct answer is $\frac{\sqrt{3}}{2}$ km.
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