Multiple choice

ABC is a vertical tower of height H. The angle of elevation of B from a point P, which is at a distance of 2a from the foot of the tower A, is 30° and the angle of elevation of C from a point Q, which is at a distance of 'a' from A, is 60°. What is the ratio of AB to BC?

  1. 1 : 1

  2. 3 : 1

  3. 1 : 2

  4. 2 : 1

  5. 1 : 3
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D Correct answer
Explanation

In triangle PAB, tan(30) = H / 2a, so H = 2a * tan(30) = 2a / sqrt(3). In triangle QAC, tan(60) = (H+BC) / a, so H+BC = a * tan(60) = a * sqrt(3). Thus, BC = a * sqrt(3) - 2a / sqrt(3) = (3a - 2a) / sqrt(3) = a / sqrt(3). The ratio AB/BC = H/BC = (2a/sqrt(3)) / (a/sqrt(3)) = 2/1.

AI explanation

Let the distance from the foot of the tower A to point B be x, so the elevation of B from P is tan 30, meaning x divided by 2a equals 1 divided by root 3, which gives x as 2a divided by root 3. The elevation of C from Q is tan 60, so H divided by a equals root 3, yielding H as a root 3. The length of AB is 2a divided by root 3, and the length of BC is H minus x, which simplifies to a root 3 minus 2a divided by root 3, equaling a divided by root 3. The ratio of AB to BC is therefore 2 to 1.

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