Measuring Along Upgrade – Forward Shift Needed with a 20 m Chain on Slope θ While measuring distance uphill with a 20 m chain, the forward (upper) end of the chain must be shifted ahead by what amount so that each stepped position represents a true 20 m horizontal measure? (θ = slope angle)

Introduction / Context:
On sloping ground, stepping with a chain is often used to measure horizontal distances. If the chain is laid along the slope, a geometric adjustment is required so that the horizontal projection equals the nominal chain length. This adjustment is implemented by advancing the forward end by a specific shift each step.

Given Data / Assumptions:

• Nominal chain length L = 20 m.
• Slope angle = θ (upgrade).
• Goal: ensure each position corresponds to a 20 m horizontal distance.

Concept / Approach:

If the chain lies along the slope, its horizontal projection equals L cos θ. To make the horizontal projection equal exactly 20 m, the actual sloping length between pegs must be 20 sec θ. Therefore, the forward end must be advanced beyond 20 m by the amount Δ = 20 sec θ − 20 = 20 (sec θ − 1). This shift ensures each step measures 20 m horizontally despite the slope.

Step-by-Step Solution:

Verification / Alternative check:

For small θ, sec θ ≈ 1 + θ^2/2 ⇒ Δ ≈ 10 θ^2 (metres), a small positive value, matching practical expectations for gentle slopes.

Why Other Options Are Wrong:

20 (cos θ − 1) is negative; 20 (sin θ − 1) and 20 (cosec θ − 1) are unrelated to horizontal projection; 20 tan θ is not the necessary shift derived from the projection requirement.

Common Pitfalls:

Forgetting that we must increase the ground step length to compensate for the slope; confusing the forward shift formula with slope reduction formulas used for converting slope distance to horizontal.

Final Answer:

20 (sec θ − 1)