Multiple choice

The upper $\dfrac{3}{4}th$ portion of a vertical pole subtends an angle $\tan^{-1}\dfrac{3}{5}$ at a point in the horizontal plane through its foot and at a distance $40\ m$ from the foot. The possible height of the vertical pole is

  1. $40\ m$
  2. $160\ m$
  3. $10\ m$
  4. $200\ m$
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B Correct answer
AI explanation

Let the total height of the pole be H. The angle subtended by the upper 3/4 portion is tan inverse of 3/5, meaning tan theta = 3/5. The elevation to the top of the pole is tan alpha = H/40, and the elevation to the bottom of the upper 3/4 portion is tan beta = H/(4 times 40) = H/160. Using the tangent difference formula, tan theta equals (tan alpha minus tan beta) divided by (1 plus tan alpha tan beta). Substituting the values gives 3/5 = (3H/160) / (1 + H squared / 6400), which simplifies to H squared minus 200H plus 6400 = 0. Factoring this quadratic equation yields (H minus 40) times (H minus 160) = 0, so the possible height is 160 m.

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