With what accuracy must a difference in elevation between two ends of a 30 m tape be known if the difference in the elevation is 3 m and the accuracy ratio is to be at least 1 in 600000 ?
- A0·0005 mCorrect
- B0·0157 m
- C0·0003 m
- D0·0006 m
Solution
Concept
Error propagation in slope correction
Formula
Ch = h² / 2L; dCh = (h / L) dh
Step-by-step solution
1. Allowable error in length = 30 / 600 000 = 5 × 10⁻⁵ m. 2. dCh = (h/L) dh = (3/30) dh = 0.1 dh. 3. dh = 5×10⁻⁵ / 0.1 = 5 × 10⁻⁴ m = 0.0005 m.
Explanation
The slope correction depends on the height difference, so a small error in h creates an error in the corrected length. Setting this error equal to the allowable error gives the required accuracy of h.
Why the other options are wrong
0.0157 m is the value for a much lower accuracy ratio (about 1 in 19 100). 0.0003 and 0.0006 m do not satisfy dh = 5×10⁻⁵ L/h.
Shortcut
dh = (allowable error × L) / h.
Common mistake
Forgetting to differentiate Ch with respect to h.
Correct answer: A) 0·0005 m