Multiple choice

From the top of a building 60m high, the angle of elevation and depression of the top and the foot of another building are α and β respectively. Find the height of the second building.

  1. 60(1+ tan α tanβ)

  2. 60(1+ cot α tanβ)

  3. 60(1+ tan α cotβ)

  4. 60(1- tan αcotβ)

  5. None
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the first building be AB (60m) and the second be CD. The distance between them is d. From the top of AB, the angle of depression to the foot of CD is beta, so tan(beta) = 60/d, d = 60 * cot(beta). The angle of elevation to the top of CD is alpha, so the height of the part of CD above the level of A is d * tan(alpha) = 60 * cot(beta) * tan(alpha). Total height = 60 + 60 * cot(beta) * tan(alpha) = 60(1 + tan(alpha) * cot(beta)).

AI explanation

The horizontal distance between the buildings is found using the angle of depression from the first building, so tan beta equals 60 divided by the distance, meaning the distance equals 60 cot beta. The extra height of the second building is found using the angle of elevation, so tan alpha equals extra height divided by the distance, giving an extra height of 60 cot beta tan alpha. The total height is 60 plus 60 tan alpha cot beta, which factors to 60(1 + tan alpha cot beta).

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