Question : Two men standing on the same side of a pillar, 75 metres high, observe the angles of elevation of the top of the pillar to be $30^{\circ}$ and $60^{\circ}$, respectively. The distance between the two men is:
Option 1: $100\sqrt{3}$ m
Option 2: $100$ m
Option 3: $50\sqrt{3}$ m
Option 4: $25\sqrt{3}$ m
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Correct Answer: $50\sqrt{3}$ m
Solution :
Let AB denote the pillar. Let C and D be the two men observing the angle of elevation of the top of the pillar as $30^{\circ}$ and $60^{\circ}$, respectively.
In $\triangle$ABD,
$\tan$ $60^\circ$ = $\frac{75}{BD}$
⇒ BD = $\frac{75}{\sqrt3}$ = $25\sqrt3$ m
In $\triangle$ABC,
$ \tan$ $60^\circ$ = $\frac{75}{BC}$
⇒ BC = $\frac{75}{\frac{1}{\sqrt3}}$ = $75\sqrt3$ m
⇒ CD = BC – BD = $75\sqrt3$ – $25\sqrt3$ = $50\sqrt3$ m
So, the distance between two men is $50\sqrt3$m.
Hence, the correct answer is $50\sqrt3$ m.
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