Depression changes from 30° to 60°: From the top of a 100 m tower, the angle of depression of a car is first 30° and later 60°. Assuming the car moves in a straight line toward the tower at constant level, find the distance the car travels between the two observations.
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A100 √3
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B200 √3 / 3
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C100 √3 / 3
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D200 √3
Answer
Correct Answer: 200 √3 / 3
Explanation
Introduction / Context:Angles of depression equal angles of elevation from the same horizontal line. We compare two right triangles formed by the car’s positions and the tower height.
Given Data / Assumptions:
- Tower height H = 100 m.
- First depression = 30°; second depression = 60°.
- Car moves horizontally toward the tower on level ground.
Concept / Approach:Horizontal distance d = H / tan(depression). The travel distance equals d1 - d2 between the two instants.
Step-by-Step Solution:
d1 = 100 / tan 30° = 100 √3.d2 = 100 / tan 60° = 100 / √3.Travel = d1 - d2 = 100 √3 - 100/√3 = (100(3 - 1))/√3 = 200/√3 = 200 √3 / 3.Verification / Alternative check:Numeric: √3 ≈ 1.732 → 200/1.732 ≈ 115.47 m. This matches 200 √3 / 3 ≈ 115.47 m.
Why Other Options Are Wrong:100 √3 and 200 √3 are too large; 100 √3 / 3 is half the correct value; only 200 √3 / 3 matches the geometry.
Common Pitfalls:Using cot instead of tan (or forgetting that depression equals elevation), subtracting in wrong order, or mixing degrees and radians in calculators.
Final Answer:200 √3 / 3