The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60^\circ. By going y metre away from that point, it becomes 30^\circ. Which one of the following relations is correct?

Explanation

Let initial distance be d. From the first position, \tan 60^\circ = \frac{x}{d} \implies d = \frac{x}{\sqrt{3}}. From the second position, \tan 30^\circ = \frac{x}{d+y} \implies d+y = x\sqrt{3}. Substituting d gives \frac{x}{\sqrt{3}} + y = x\sqrt{3}, which simplifies to y = x\sqrt{3} - \frac{x}{\sqrt{3}} = \frac{2x}{\sqrt{3}}. Thus, 2x = \sqrt{3}y.

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