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From a horizontal distance of 50 m, the angles of elevation of the top and the bottom of a vertical cliff face are \( \Large 45^{\circ} \) and \( \Large 30^{\circ} \) respectively. The height of the cliff face (in m) is

A) \( \Large \frac{50}{\sqrt{3}} \)
B) \( \Large \frac{50}{\sqrt{2}} \)
C) \( \Large \frac{50}{2\sqrt{3}} \)
D) \( \Large 50 (1 - \frac{1}{\sqrt{3}}) \)

Correct Answer:




D) \( \Large 50 (1 - \frac{1}{\sqrt{3}}) \)

Description for Correct answer:
Let the height of cliff PR is h m.

\( \Large \therefore \tan 30^{\circ} = \frac{x}{50} \)

\( \Large \frac{1}{\sqrt{3}} = \frac{x}{50} \Rightarrow x = \frac{50}{\sqrt{3}} \) ....(i)

And \( \Large \tan 45^{\circ} = \frac{(x + h)}{50} \)

=> x + h = 50 ....(ii)

\( \Large \therefore \ h = 50 - \frac{50}{\sqrt{3}} = 50 (1 - \frac{1}{\sqrt{3}}) \) ( By (i) & (ii) )

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