As observed from the top of a lighthouse, 54 m high above the sea-level, the…
2020
As observed from the top of a lighthouse, 54 m high above the sea-level, the angle of depression of a ship, sailing directly towards it, changes from 45° to 60°. The distance (in m) travelled by the ship during the period of observation is (correct to one decimal place):
- A.
22.8
- B.
37.8
- C.
27.5
- D.
39.5
Show answer & explanation
Correct answer: A
CONCEPT
For an angle of depression/elevation θ, the vertical height, horizontal distance, and angle form a right triangle. The tangent ratio is tan θ = opposite side / adjacent side, so horizontal distance = height / tan θ.
When an object moves directly toward the foot of the vertical height, the distance travelled is the decrease in its horizontal distance from that foot.
APPLICATION
Let the lighthouse height be h = 54 m.
At 45°, the horizontal distance from the lighthouse base is d1 = 54 / tan 45° = 54 / 1 = 54 m.
At 60°, the horizontal distance is d2 = 54 / tan 60° = 54 / √3 ≈ 31.1769 m.
The ship travelled d1 − d2 = 54 − 31.1769 ≈ 22.8231 m.
Rounded to one decimal place, the distance is 22.8 m.
CROSS-CHECK
The 60° sighting must be closer than the 45° sighting because the angle is steeper. The computed distances, 31.1769 m and 54 m, satisfy that order, and their difference gives the travelled distance.
Result: 22.8 m.