The correction to be applied to each 30-meter chain length along a θ° slope is 

📐 Understanding the Premise

This question asks for the "correction" in a specific way. Instead of the typical method where you measure along the slope and subtract a correction to find the horizontal distance, this question's structure implies finding the extra length you measure along the slope compared to the horizontal distance.

Here, we assume the '30-meter chain length' refers to the desired horizontal distance (H), and we need to find how much longer the slope distance (L) is.

🔬 Derivation of the Formula

Let's visualize the scenario as a right-angled triangle:

• AC is the slope distance (L), which is the hypotenuse.
• AB is the horizontal distance (H = 30 m), the adjacent side.
• θ is the angle of the slope at point A.

Step 1: Relate the sides using trigonometry

The relationship between the adjacent side (H), the hypotenuse (L), and the angle (θ) is given by the cosine function:

$$ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{H}{L} $$

Step 2: Solve for the slope distance (L)

We need to find the length along the slope (L). Rearranging the formula:

$$ L = \frac{H}{\cos(\theta)} $$

Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we can write:

$$ L = H \cdot \sec(\theta) $$

Step 3: Calculate the Correction (C)

The correction (C) is the difference between the longer slope distance (L) and the shorter horizontal distance (H).

$$ C = L - H $$

Now, substitute the expression for L from Step 2:

$$ C = (H \cdot \sec(\theta)) - H $$

Step 4: Factor and Finalize the Formula

Factor out H to get the general formula:

$$ C = H(\sec(\theta) - 1) $$

Given that the horizontal length H is 30 m, we get:

$$ C = 30(\sec(\theta) - 1) \text{ m} $$