The correction per chain length of 100 links along a slope of α° is

📐 Understanding Slope Correction

When measuring distance along a slope (L), the measurement is always longer than the true horizontal distance (H). To get the correct map distance, we must apply a slope correction (C), which is always negative (subtractive).

The fundamental relationship is: Horizontal Distance (H) = Slope Distance (L) - Correction (C). The correction itself is the difference: \( C = L - H \).

🔬 Derivation of the Approximate Formula

The options provided are based on an empirical or approximate formula for small angles measured in degrees. Let's derive it from fundamental principles.

Step 1: The Exact Trigonometric Formula

From the right-angled triangle formed by the slope, we know \( H = L \cos(\alpha) \). Therefore, the exact correction is:

$$ C = L - H = L - L \cos(\alpha) = L(1 - \cos(\alpha)) $$

Step 2: Small Angle Approximation (for Degrees)

For small angles, the Taylor series expansion for cosine is \( \cos(x) \approx 1 - x^2/2 \), where x is in radians. To use our angle \( \alpha^\circ \) (in degrees), we must first convert it to radians:

$$ \alpha_{rad} = \alpha^\circ \times \frac{\pi}{180} $$

Substituting this into the cosine approximation:

$$ \cos(\alpha^\circ) \approx 1 - \frac{(\alpha^\circ \times \frac{\pi}{180})^2}{2} $$

Step 3: Substitute Approximation into Correction Formula

Now we put this back into our correction formula from Step 1:

$$ C \approx L \left( 1 - \left(1 - \frac{(\alpha^\circ \frac{\pi}{180})^2}{2}\right) \right) $$

$$ C \approx L \left( \frac{(\alpha^\circ)^2 \pi^2}{2 \times 180^2} \right) $$

Calculating the constant term: \( \frac{\pi^2}{2 \times 180^2} \approx \frac{9.8696}{64800} \approx 0.0001523 \)

So, the precise approximate formula is: $$ C \approx L \times (\alpha^\circ)^2 \times 0.0001523 $$

Step 4: Apply to a 100-Link Chain

For a chain length of L = 100 links, the correction (in links) is:

$$ C \approx 100 \times (\alpha^\circ)^2 \times 0.0001523 \approx 0.01523 (\alpha^\circ)^2 $$

Step 5: Connecting to the Answer

Now let's look at the formula from option A:

$$ \frac{1.5\alpha^2}{100} = 0.015\alpha^2 $$

As you can see, the value 0.015 is a convenient rounding of the more precise constant 0.01523. For fieldwork and exam purposes, this empirical formula is often used for its simplicity.