Multiple choice

The angle of elevation of the top of a tower from a point on the ground at some distance from its base is 60°. The angle of elevation of the top of the tower from a point 20 m above the same point on the ground is 30°. What is the height of the tower?

  1. 30 m

  2. 20 m

  3. 15 m

  4. 10 m

  5. None of the above
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A Correct answer
Explanation

Let h be height, d be distance. tan(60) = h/d => d = h/sqrt(3). tan(30) = (h-20)/d => d = (h-20) * sqrt(3). Equating: h/sqrt(3) = (h-20) * sqrt(3) => h = 3(h-20) => h = 3h - 60 => 2h = 60 => h = 30.

AI explanation

Let the distance from the base to the first point be D, then tan(60 degrees) equals the height of the tower divided by D, meaning D equals the height divided by the square root of 3. Using the second point 20 meters above the ground gives tan(30 degrees) equals the height minus 20 divided by D, and substituting D gives 1 divided by the square root of 3 equals the height minus 20 multiplied by the square root of 3 divided by the height. Solving this equation results in a height of 30 meters.

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