Square plot, pole at D — mixed observation points ABCD is a square. The angle of elevation of the top of a pole at D is 30° as seen from A and also from C. From B, the angle of elevation is Θ. Find tan Θ.
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A√6
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B1/√6
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C√3/√2
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D√2/√3
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E—
Answer
Correct Answer: 1/√6
Explanation
Introduction / Context:In a square, distances from D to adjacent corners (A and C) equal the side length s, while distance from D to the opposite corner (B) equals the diagonal s√2. Using the common pole height derived from the 30° observations at A and C, we can find tan Θ at B.
Given Data / Assumptions:
- AD = CD = s; BD = s√2.
- tan 30° = height/AD ⇒ height h = s/√3.
- We seek tan Θ = h/BD.
Concept / Approach:Use the 30° observation to compute h in terms of the side s, then evaluate tan Θ from the more distant point B using the diagonal baseline BD.
Step-by-Step Solution:
h = s * tan 30° = s / √3BD = s√2tan Θ = h / BD = (s/√3) / (s√2) = 1/√6Verification / Alternative check:If s = √6 (arbitrary scaling), then h = 1 and BD = √12; tan Θ = 1/√12 = 1/ (2*√3) = 1/√6 after simplification consistency check.
Why Other Options Are Wrong:√6 or √3/√2 imply much larger angles inconsistent with the geometry; √2/√3 is also incorrect given h and BD.
Common Pitfalls:Treating BD as s instead of s√2, or mixing up sine and tangent when computing h.
Final Answer:1/√6