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A concrete pillar is 30 feet. high and has a flag staff at its top. The pillar and the flag staff subtend equal angles at a point distant 50 feet from the foot of the tower. Height of the flag staff in feet is

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answer is 64.

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Detailed Solution

$\frac{b}{a} = \tan θ, \frac{b + x}{a} = \tan 2 θ$ or $\frac{b + x}{a} = \frac{2 \tan θ}{1 - \tan^{2} θ} = \frac{2 \cdot \frac{b}{a}}{1 - \frac{b^{2}}{a^{2}}}$ $\frac{b + x}{a} = \frac{2 a b}{a^{2} - b^{2}} ∴ x = \frac{2 a^{2} b}{a^{2} - b^{2}} - b$

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or $x = \frac{a^{2} b + b^{3}}{a^{2} - b^{2}} = \frac{b (a^{2} + b^{2})}{a^{2} - b^{2}}$

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