Multiple choice

From the top of a building $15$ m high the angle of elevation of the top a tower is found to be $30^0$. From the bottom of the same building, the angle of elevation of the tower id found to be $60^0$. Find the height of the tower and the distance between the tower and the building.

  1. $h= 22.5 , D=38.97$
  2. $h=22.5, D=12.97$
  3. $h= 38.97, D=22.5$
  4. $h=12.97, D= 22.5$
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B Correct answer
Explanation

Let D be the horizontal distance and h the tower height. From the two elevations, tan 60 degrees = h/D and tan 30 degrees = (h - 15)/D, which gives D = 15sqrt(3)/2 ≈ 12.97 m and h = 22.5 m.

AI explanation

Let the horizontal distance between the building and the tower be x, making the height difference between the tower and the building equal to x tan(30 degrees), which is x divided by the square root of 3. The total height of the tower is 15 plus x divided by the square root of 3, and from the bottom of the building this height also equals x tan(60 degrees), or x times the square root of 3. Equating the two expressions for the tower height gives 15 equals 2x divided by the square root of 3, yielding a distance x of 12.99 meters. Substituting x back into the formula gives the tower height as 22.5 meters, so the height is 22.5 and the distance is 12.97.

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