From the top of a building 60 metres high, the angles of depression of the top…
2024
From the top of a building 60 metres high, the angles of depression of the top and bottom of a tower are 30° and 60° respectively. What is the height of the tower?
- A.
40 m
- B.
30 m
- C.
20 m
- D.
50 m
Show answer & explanation
Correct answer: A
When one observer stands above two objects, the angle of depression to each object equals the angle of elevation from that object back to the observer (alternate angles, since the line of sight and the horizontal are parallel to the ground). The tangent of that angle then equals the vertical height above the object divided by the horizontal distance between the observer and the object.
Applying this to the building and tower:
Let the horizontal distance between the building and the tower be d. The angle of depression to the bottom of the tower is 60°, so tan 60° = 60/d, giving d = 60/√3 = 20√3 m.
The angle of depression to the top of the tower is 30°. Let the vertical gap between the top of the building and the top of the tower be x, so tan 30° = x/d, giving x = 20√3 × (1/√3) = 20 m.
The tower's height equals the building's height minus this vertical gap: 60 − 20 = 40 m.
Cross-check: if the tower is 40 m tall, the gap between the two tops is 60 − 40 = 20 m, and 20/d = 20/(20√3) = 1/√3 = tan 30°, which matches the given angle of depression to the tower's top — confirming the result.