Small-angle facts and error growth in traverses: Which of the following statements are correct regarding angular/linear relations and error propagation?
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A1 second of arc corresponds to a displacement ratio of about 1:206,300
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B1 degree of arc corresponds to a displacement ratio of about 1:57
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CAngular errors along a traverse tend to propagate approximately with the square root of the number of stations
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DErrors from linear measurements tend to be roughly proportional to the lengths of the lines
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EAll of the above
Answer
Correct Answer: All of the above
Explanation
Introduction / Context:Survey accuracy depends on both angular and linear measurements. Small-angle approximations and empirical error growth rules guide specifications, checking, and adjustment strategies in traversing and triangulation.
Given Data / Assumptions:
- Using standard small-angle relations in radians.
- Random errors assumed independent and unbiased.
- Practical field conditions with conventional instruments.
Concept / Approach:One radian equals about 57.3 degrees, hence 1 degree corresponds to ~1/57 rad. One arc-second equals about 1/206,265 rad, giving the 1:206,300 displacement ratio for small deflections. Random angular misclosures grow with sqrt(n) where n is the number of stations, while linear random errors tend to scale with total length measured, reflecting accumulation of many small unbiased contributions.
Step-by-Step Solution:
Translate degrees/seconds to radian-based displacement ratios.Relate random error accumulation to sqrt(n) growth (root-sum-square).Recognize linear error scaling with total measured length.Synthesize: each statement A–D is a standard rule of thumb; thus E is correct.Verification / Alternative check:Bowditch (compass) rule and least-squares theory corroborate sqrt(n) angular propagation and length-proportional linear behavior in typical field work.
Why Other Options Are Wrong:Any single statement alone is incomplete; the bundle of all four best represents accepted surveying practice.
Common Pitfalls:Confusing systematic scale errors with random errors; mixing degree–radian conversions.
Final Answer:All of the above