Multiple choice

At a certain point the angle of elevation of a tower is found to be $\cot^{-1}\dfrac35$. On walking $32\ m$ directly towards the tower its angle of elevation is $\cot^{-1}\dfrac25$ . The height of the tower (in metres) is

  1. $32$
  2. $160$
  3. $320$
  4. $340$
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B Correct answer
Explanation

Let h be height, x be initial distance. cot(theta1) = x/h = 3/5, so x = 3h/5. After walking 32m, distance is x-32. cot(theta2) = (x-32)/h = 2/5, so x-32 = 2h/5. Substituting x: 3h/5 - 2h/5 = 32. h/5 = 32, so h = 160.

AI explanation

Let the tower height be h and the initial distance from the tower be x. Since cot inverse of (3/5) equals tan inverse of (5/3), we write tan of the first angle as h/x = 5/3, or x = 3h/5. After walking 32 m closer, the new distance is x - 32, and tan of the second angle is h/(x - 32) = 5/2, which gives x - 32 = 2h/5. Substituting x = 3h/5 into the second equation yields 3h/5 - 32 = 2h/5, so h/5 = 32 and h = 160.

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