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(C) Let the total height of the pole be $H$. The lower part is $\frac{H}{3}$ and the upper part is $\frac{2H}{3}$.Let the observer be at point $O$ at

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$10\sqrt{3}$

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(C) Let the total height of the pole be $H$. The lower part is $\frac{H}{3}$ and the upper part is $\frac{2H}{3}$.Let the observer be at point $O$ at a distance of $10 \ m$ from the base of the pole.Let the angle subtended by the lower part be $\alpha$ and the angle subtended by the upper part be $\alpha$.From the geometry,$	an \alpha = \frac{H/3}{10} = \frac{H}{30}$.The total angle subtended by the whole pole is $2\alpha$.Thus,$	an(2\alpha) = \frac{H}{10}$.Using the formula $	an(2\alpha) = \frac{2	an \alpha}{1 - 	an^2 \alpha}$,we have:$\frac{H}{10} = \frac{2(H/30)}{1 - (H/30)^2}$$1 = \frac{2/3}{1 - H^2/900}$$1 - \frac{H^2}{900} = \frac{2}{3}$$\frac{H^2}{900} = 1 - \frac{2}{3} = \frac{1}{3}$$H^2 = 300$$H = \sqrt{300} = 10\sqrt{3} \ m$.

$A$ person standing at a distance of $10 \ m$ from a pole observes that the angle subtended by the lower $\frac{1}{3}$rd part and the upper $\frac{2}{3}$rd part of the pole are the same. Find the height of the pole.

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