Stepping on Uniform Slope – Approximate Error Magnitude A uniform slope between two points was measured by the stepping method. If the vertical difference in level is 1.8 m and the single-slope (hypotenuse) distance is 15 m, what is the approximate error magnitude attributable to using the slope length instead of the true horizontal distance?

Introduction:
On sloping ground, horizontal distance H is less than the corresponding slope distance s. If one mistakenly uses s (or fails to reduce a slope tape length to horizontal), a systematic error is introduced. This question estimates that error for a simple case and asks for its sign and nature.

Given Data / Assumptions:

• Slope distance s = 15 m (single shot between two points).
• Difference in level h = 1.8 m (uniform slope).
• True horizontal distance H = √(s^2 − h^2).

Concept / Approach:

For a right triangle with hypotenuse s and vertical leg h, the horizontal projection is H. If stepping is done correctly with a level tape, H is obtained. If, however, one inadvertently uses s as though it were horizontal, the measured length is too great. The error relative to the true horizontal is H − s (a negative quantity), which numerically equals −(s − H).

Step-by-Step Solution:

Verification / Alternative check:

Small-angle approximation: for small slopes, s − H ≈ h^2 / (2s) = (1.8^2)/(30) ≈ 0.108 m, matching the calculation.

Why Other Options Are Wrong:

'+ 0.11 m' has the wrong sign; 'compensating' is incorrect because the bias repeats with each similarly sloped segment; larger values like 0.50 m do not fit the geometry.

Common Pitfalls:

Confusing whether to subtract H from s or vice versa; forgetting to square h in the Pythagorean relation.

Final Answer:

Cumulative, − 0.11 m