Multiple choice

A man standing on a horizontal plane, observes the angle of elevation of the top of a tower to be $\alpha$. After walking a distance equal to double the height of the tower, the angle of the elevation becomes $2\alpha$, then $\alpha$ is equal to

  1. $\dfrac { \pi }{ 2 } $
  2. $\dfrac { \pi }{ 6 } $
  3. $\dfrac { \pi }{ 12 } $
  4. $\dfrac { \pi }{ 18 } $
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C Correct answer
Explanation

Let tower height be h. tan(alpha) = h/x and tan(2*alpha) = h/(x-2h). Substituting x = h/tan(alpha) into the second equation: tan(2*alpha) = h / (h/tan(alpha) - 2h) = tan(alpha) / (1 - 2*tan(alpha)). Using tan(2*alpha) = 2*tan(alpha) / (1 - tan^2(alpha)), we solve for tan(alpha).

AI explanation

Let the tower height be h, making the initial distance 2h * cot(alpha). After walking a distance equal to double the tower height, the new distance is 2h * cot(alpha) - 2h. In the new right triangle, tan(2 * alpha) = h / (2h * cot(alpha) - 2h). Using the double angle identity tan(2 * alpha) = 2 * tan(alpha) / (1 - tan^2(alpha)) and simplifying yields tan(alpha) = 2 + sqrt(3). Since tan(15 degrees) = tan(pi / 12) equals 2 + sqrt(3), alpha is pi / 12.

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