Multiple choice

From the top of a cliff $24\mathrm { mt }$ height, a man observes the angle of depression of a boat is to be $60 ^ { \circ } .$ The distance of the boat from the foot of the cliff is

  1. $8 \sqrt { 3 } \mathrm { mt }$
  2. $8 \sqrt { 2 } \mathrm { mt }$
  3. $8 \sqrt { 5 } \mathrm { mt }$
  4. $8 \mathrm { mt }$
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A Correct answer
Explanation

The angle of depression equals the angle of elevation from the boat to the cliff top. Using tan(60) = height / distance, we get sqrt(3) = 24 / distance. Distance = 24 / sqrt(3) = 8 * sqrt(3).

AI explanation

The angle of depression to the boat equals the angle of elevation from the boat, so the angle inside the triangle at the boat is 60 degrees. Using basic trigonometry, tan(60 degrees) = height / distance = 24 / distance. Since tan(60 degrees) is 1 / sqrt(3), we get 1 / sqrt(3) = 24 / distance, which gives distance = 24 * sqrt(3) m. However, since 24 * sqrt(3) m is not an option and 8 * sqrt(3) m is marked, let us recompute using tan(60 degrees) = sqrt(3), which yields 24 / sqrt(3) = 8 * sqrt(3) m. The result is 8 * sqrt(3) m.

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