Slope correction in chaining: For end points with level difference h over a measured slope length l, the exact horizontal distance is √(l² − h²). Expanding in series gives the slope correction c ≈ (h² / 2l) + (h⁴ / 8l³) + … . The second term may be neglected if h over a 20 m distance is less than approximately what value?

Introduction / Context:

In chaining along a slope, the measured length l exceeds the horizontal distance D. Converting the sloping measurement to horizontal requires the slope correction. A series expansion allows us to judge when higher-order terms can be ignored without introducing significant error—a key consideration in ordinary engineering surveys.

Given Data / Assumptions:

• Slope length l = 20 m (chain length or taped segment).
• Difference in level between the two ends = h.
• Horizontal distance D = √(l² − h²); slope correction c = l − D.
• Acceptable drafting/measurement error ~1 mm to a few mm.

Concept / Approach:

Binomial expansion: √(l² − h²) = l √(1 − (h²/l²)) ≈ l [1 − (h²)/(2l²) − (h⁴)/(8l⁴) …]. Thus c = l − D ≈ (h²)/(2l) + (h⁴)/(8l³) + … . The question seeks the threshold h below which the second term is negligible for l = 20 m.

Step-by-Step Solution:

Verification / Alternative check:

Compare the first term for h = 3 m: (h²)/(2l) = 9 / 40 = 0.225 m; the second term is ~0.0013 m, i.e., <1% of the first term and within plotting tolerances.

Why Other Options Are Wrong:

• 2 m, 1 m, 0.5 m, 0.25 m: all are acceptable but overly conservative; the question asks the value below which neglect is justified—3 m is the accepted classroom threshold for a 20 m length.

Common Pitfalls:

• Using c ≈ l − l cos θ directly without assessing higher-order terms for steep slopes.
• Confusing slope length l with horizontal D when computing areas.

Final Answer:

3 m