Tacheometry – inclined sights correction: For horizontal distance by stadia with line of sight inclined at angle θ (staff vertical), the standard horizontal-sight formula can be adapted by multiplying which constants?
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AMultiply both multiplying and additive constants by sin^2 θ
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BMultiply both multiplying and additive constants by cos^2 θ
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CMultiply both constants by cos θ
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DMultiply both constants by sin θ
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EMultiply the multiplying constant by cos^2 θ and the additive constant by cos θ
Answer
Correct Answer: Multiply the multiplying constant by cos^2 θ and the additive constant by cos θ
Explanation
Introduction / Context:In stadia (tacheometric) surveying with a vertical staff, the familiar horizontal-sight relation is D = k * s + c, where D is horizontal distance, s is staff intercept, k is the multiplying constant, and c is the additive constant. When sights are inclined by angle θ, appropriate trigonometric factors modify these constants to recover horizontal distance directly.
Given Data / Assumptions:
- Staff is held vertical; line of sight makes angle θ with the horizontal.
- k and c are instrument constants determined by focusing system and stadia interval.
- We seek the formula for horizontal distance (not slope distance).
Concept / Approach:For inclined sights with vertical staff, the slope-distance relation is S = k * s + c. The horizontal component is D = S * cos θ, but the staff intercept seen by the inclined line of sight also projects with a cos θ factor. Combining these effects yields the well-known result D = k * s * cos^2 θ + c * cos θ, which corresponds to multiplying k by cos^2 θ and c by cos θ in the horizontal-sight formula.
Step-by-Step Solution:
Start from slope distance: S = k * s + c.Project to horizontal: D = S * cos θ = (k * s + c) * cos θ.Account for vertical staff intercept under inclination: effective intercept contributing to S already scales with cos θ, producing D = k * s * cos^2 θ + c * cos θ.Hence, adapt constants: k → k * cos^2 θ; c → c * cos θ.Verification / Alternative check:Textbook derivations using geometry of similar triangles for the stadia diaphragm with vertical staff confirm the cos^2 θ and cos θ factors for horizontal distance.
Why Other Options Are Wrong:
- sin θ or sin^2 θ factors apply to vertical component computations, not horizontal distance.
- Using a single cos θ for both constants underestimates horizontal distance.
- Applying cos^2 θ to both constants overcorrects the additive part.
Common Pitfalls:Confusing formulas for staff held normal to the line of sight; forgetting to distinguish slope distance from horizontal distance; mixing up the roles of k and c.
Final Answer:Multiply the multiplying constant by cos^2 θ and the additive constant by cos θ