Correction per chain length of 100 links along a slope having a rise of 1 unit in n horizontal units is:
The source material provided answer 'A' (\(100/n^2\)), which is based on a less common approximation. See the detailed derivation below.
📈 Understanding Slope as a Gradient
A slope described as a "rise of 1 unit in n horizontal units" defines the gradient of the terrain. This forms a right-angled triangle where:
- The vertical rise (h) = 1 unit
- The horizontal distance (H) = n units
- The slope distance (L) is the hypotenuse.
The goal of the correction is to find the difference between the measured slope distance (L) and the true horizontal distance (H).
🔬 Derivation of the Slope Correction Formula
We can derive the formula using the Pythagorean theorem and a binomial approximation for small slopes.
Step 1: Relate the distances
From the right-angled triangle, we have: $$L^2 = H^2 + h^2$$
Step 2: Express L in terms of H
$$L = \sqrt{H^2 + h^2} = H \sqrt{1 + \frac{h^2}{H^2}}$$
Step 3: Apply Binomial Approximation
For small slopes, the term \(h^2/H^2\) is small. We can use the approximation \((1+x)^k \approx 1+kx\):
$$L \approx H \left(1 + \frac{1}{2} \frac{h^2}{H^2}\right) = H + \frac{h^2}{2H}$$
Step 4: Calculate the Correction
The correction C is the difference \(L-H\). Note that the correction is always subtracted, so we consider its magnitude.
$$C = L - H \approx \frac{h^2}{2H}$$
Step 5: Substitute the Gradient
We are given the gradient \( \frac{h}{H} = \frac{1}{n} \). For a full chain length \(L\), we approximate \(H \approx L\). The total rise over this length is \(h = L/n\).
Substitute these into the correction formula:
$$C \approx \frac{(L/n)^2}{2L} = \frac{L^2/n^2}{2L} = \frac{L}{2n^2}$$
Step 6: Apply for a 100-Link Chain
For a chain where L = 100 links:
Correction \( C \approx \frac{100}{2n^2} = \frac{50}{n^2} \)
💡 Analysis of the Options
The standard, mathematically-derived formula for this correction is \( \mathbf{50/n^2} \), which corresponds to option E. This is the most accurate formula based on the binomial approximation for small slopes.
The answer provided in the source material was option A, \( \mathbf{100/n^2} \). This value likely comes from a different, less accurate approximation or a common simplification sometimes found in textbooks. For a correct conceptual understanding and for most exam scenarios, the formula derived (\( \mathbf{50/n^2} \)) is the one to remember.