Multiple choice

From the top of a cliff $200$ metres high, the angles of depression of the top and bottom of a tower are observed to be ${30}^{o}$ and ${60}^{o}$. Find the height of the tower and calculate the distance between them.

  1. Height:$=156$; Distance$=119.7m$
  2. Height$=133\cfrac{1}{3}$; Distance$=115.46m$
  3. Height$=220$; Distance$=112.76m$

  4. None of these
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B Correct answer
Explanation

Cliff height = 200. Let tower height be h and distance be d. From top of cliff to tower top: tan(30) = (200 - h) / d. From top of cliff to tower bottom: tan(60) = 200 / d. d = 200 / sqrt(3) = 115.47m. 200 - h = d * tan(30) = (200 / sqrt(3)) * (1 / sqrt(3)) = 200 / 3 = 66.67. h = 200 - 66.67 = 133.33m.

AI explanation

From the top of the 200 m cliff, the angle of depression to the bottom of the tower is 60 degrees, making the angle of elevation from the tower base to the cliff top 60 degrees. Using the tangent ratio, tan 60 degrees equals 200 divided by the distance between them, so the distance is 200 divided by root 3, which approximates to 115.46 m. The angle of depression to the top of the tower is 30 degrees, meaning the difference in height between the cliff and the tower, let us call it y, satisfies tan 30 degrees equals y divided by 115.46. Thus, y is 115.46 divided by root 3, which equals 200/3 m or 66.67 m. The height of the tower is the cliff height minus y, giving 200 minus 66.67, which equals 133 1/3 m.

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