From the midpoint of the line segment joining the feet of two vertical towers, the angles of elevation to their tops are 60° and 30°, respectively. Find the ratio of the heights of the two towers.
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A2 : 1
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B√3 : 1
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C3 : 2
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D3 : 1
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E1 : √3
Answer
Correct Answer: 3 : 1
Explanation
Introduction / Context:With observation at the midpoint between the bases, the horizontal distances to the two towers are equal. Thus heights are proportional to tan of their elevation angles at the same base distance.
Given Data / Assumptions:
- Equal horizontal distance d to both towers.
- Angles: 60° for the taller, 30° for the shorter (w.l.o.g.).
Concept / Approach:h ∝ tan θ when horizontal distance is constant. Therefore, ratio of heights = tan 60° : tan 30° = √3 : (1/√3) = 3 : 1.
Step-by-Step Solution:
h1 = d * tan 60° = d√3; h2 = d * tan 30° = d/√3.h1 : h2 = (d√3) : (d/√3) = 3 : 1.Verification / Alternative check:If both angles were 45°, the ratio would be 1 : 1. Here, 60° to one tower ensures it is three times the other for the same d.
Why Other Options Are Wrong:2 : 1 or √3 : 1 ignore tan 30° = 1/√3; 3 : 2 understates the difference.
Common Pitfalls:Using sin/cos instead of tan; assuming unequal distances when “midpoint” guarantees equality.
Final Answer:3 : 1