Multiple choice

There are two upright poles of height 20√3 m and 30√3 m respectively separated by a distance of 40 m. A man is standing between the two poles. What is the probability that the angle elevation of the top of the poles as seen by the man is at least 60°?

  1. 1/8

  2. 1/5

  3. 1/3

  4. 1/4
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D Correct answer
Explanation

Let the man be at distance x from the first pole (20sqrt(3)m). The angle of elevation is >= 60 degrees if tan(theta) = h/d >= sqrt(3). For the first pole, x <= 20sqrt(3)/sqrt(3) = 20. For the second pole, (40-x) <= 30sqrt(3)/sqrt(3) = 30. Thus, x <= 20 and x >= 10. The favorable range for x is [10, 20], which has length 10. The total distance is 40, so probability is 10/40 = 1/4.

AI explanation

Let the man stand at a distance x from the first pole of height 20 times the square root of 3, making his distance from the second pole 40 minus x. For the first pole, the maximum distance for a 60 degree angle is given by tan 60 degrees equals 20 times the square root of 3 divided by x, which yields x equals 20. For the second pole, the condition is tan 60 degrees equals 30 times the square root of 3 divided by the quantity 40 minus x, giving 40 minus x equals 30, or x equals 10. The successful range for x is between 0 and 20, and for the second condition it is between 10 and 40; since both conditions must be met, x must be between 10 and 20, which is a length of 10 out of the total 40 m. The probability is therefore 10 divided by 40, which is 1/4.

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