Angle of depression from 50 m lighthouse — horizontal distance From the top of a lighthouse 50 m above sea level, the angle of depression of an approaching boat is 30°. How far is the boat from the lighthouse foot along the sea surface?
Aptitude
Height and Distance
Difficulty: Easy
Choose an option
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A25 √3 m
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B25 / √3 m
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C50 √3 m
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D50 / √3 m
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E—
Answer
Correct Answer: 50 √3 m
Explanation
Introduction / Context:Angle of depression equals the angle of elevation from the boat. With known vertical height, the horizontal distance follows from the tangent relation.
Given Data / Assumptions:
- Height h = 50 m.
- Angle of depression θ = 30° ⇒ tan θ = h/d.
- Calm sea assumed; straight line of sight.
Concept / Approach:Let d be the horizontal distance. Then tan 30° = h/d ⇒ d = h / tan 30°.
Step-by-Step Solution:
tan 30° = 1/√3d = 50 / (1/√3) = 50√3 mVerification / Alternative check:Numerically, √3 ≈ 1.732 ⇒ d ≈ 86.6 m, plausible for a 50 m elevation at 30°.
Why Other Options Are Wrong:25√3 and 25/√3 correspond to halving the height; 50/√3 would be tan 60°, not tan 30°.
Common Pitfalls:Inverting the tan ratio or confusing angle of depression with angle of elevation magnitude.
Final Answer:50 √3 m