Multiple choice

A vertical pole consists of two parts, the lower part being one third of the whole. At a point in the horizontal plane through the base of the pole and distance 20m from it, the upper part of the pole subtends an angle whose tangent is 1/2. The possible heights of the pole are -

  1. $20m$ and $20\sqrt{3}$m
  2. $20m$ and $60m$
  3. $16m$ and $48m$

  4. None of these
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B Correct answer
Explanation

Let the height of the pole be H. The lower part is H/3 and the upper part is 2H/3. Let the distance from the base be 20m. The angle of elevation to the top is alpha and to the top of the lower part is beta. We are given tan(alpha - beta) = 1/2. Using the tangent subtraction formula, (tan(alpha) - tan(beta)) / (1 + tan(alpha)tan(beta)) = 1/2, where tan(alpha) = H/20 and tan(beta) = (H/3)/20 = H/60. Solving this quadratic equation for H yields 20m and 60m.

AI explanation

Let the total height of the pole be H, so the lower part is H/3 and the upper part is 2H/3. The tangent of the angle subtended by the upper part at the distance of 20 m is given by (20 * (2H/3)) / (400 + (H/3)(H)), which simplifies to 40H / (1200 + H^2). Setting this equal to 1/2 gives the quadratic equation H^2 - 80H + 1200 = 0, which factors into (H - 20)(H - 60) = 0. Therefore, the possible heights of the pole are 20 m and 60 m.

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