Multiple choice

The horizontal distance between two towers is $60\ m$ and angular depression of the top of the first as seen from the second, which is $150\ m$ in height, is $30^{0}$. The height of the first tower is

  1. $(150+20\sqrt{3})\ m$
  2. $(150+15\sqrt{3})\ m$
  3. $(150-20\sqrt{5})\ m$
  4. $(150-20\sqrt{3})\ m$
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D Correct answer
Explanation

Let h1 be the height of the first tower and h2 = 150 be the height of the second. The angle of depression is 30 degrees. tan(30) = (150 - h1) / 60. 1/sqrt(3) = (150 - h1) / 60. 60/sqrt(3) = 150 - h1. 20*sqrt(3) = 150 - h1. h1 = 150 - 20*sqrt(3).

AI explanation

The difference in height between the two towers forms the opposite side of a right triangle where the adjacent side is the horizontal distance of 60 meters. Using the tangent of the angle of depression, tan(30) = (150 - h) / 60. Since tan(30) is 1 / sqrt(3), we have (150 - h) = 60 / sqrt(3), which rationalizes to 20 * sqrt(3). Solving for h gives h = 150 - 20 * sqrt(3). The result is (150 - 20 * sqrt(3)) m.

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