The Ramp Compliance Check
A building inspector measures a wheelchair ramp serving a public entrance. The ramp rises 1.2 meters over a horizontal run of 14.4 meters. Code requires the slope angle to be no more than 4.8 degrees from horizontal, and separately requires a landing every 9 meters of horizontal run. The inspector's laser level reads the vertical rise as 1.2 m with a stated accuracy of ±0.03 m, and the run as 14.4 m with accuracy ±0.1 m. The contractor argues the ramp 'passes because 1.2 divided by 14.4 is only 0.083, well under the 1-in-12 rule of 0.0833.' The inspector is unsure whether the 1-in-12 rule and the 4.8-degree rule agree.
1. Compute the ramp's slope angle from the measurements, propagate the uncertainty, and determine whether the 4.8-degree limit is met. Then check whether the 1-in-12 rule and the 4.8-degree rule are consistent.
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Angle: tan θ = rise/run = 1.2/14.4 = 0.08333, so θ = arctan(0.08333) ≈ 4.76°.
Uncertainty: The ratio r = rise/run. Relative uncertainty ≈ sqrt((0.03/1.2)² + (0.1/14.4)²) = sqrt(0.000625 + 0.0000482) ≈ sqrt(0.000673) ≈ 0.0259, about 2.6%. So r ranges roughly 0.0812 to 0.0855. Converting: arctan(0.0812) ≈ 4.64°, arctan(0.0855) ≈ 4.89°.
Verdict: The measured angle 4.76° is below 4.8°, but the upper uncertainty bound (4.89°) exceeds it. The ramp is marginally compliant - within tolerance but not confidently so.
Rule consistency: 1-in-12 means tan θ = 1/12 = 0.08333, which is exactly 4.76°. The 4.8° rule is a rounded version of the same standard (4.76° rounds to 4.8°). They agree to within rounding. The contractor's arithmetic is right but the conclusion 'well under' is wrong: 0.083 is essentially at the limit, not comfortably below it.
2. The landing rule requires a landing every 9 meters of horizontal run. Evaluate compliance and explain how measurement uncertainty affects the judgment.
Read the worked answer
Landing count: With a 14.4 m run, the ramp needs landings at intervals no greater than 9 m. One intermediate landing at, say, 7.2 m would split the run into 7.2 m and 7.2 m, both under 9 m - compliant. If no intermediate landing exists, the single 14.4 m span violates the rule outright, since 14.4 > 9.
Uncertainty effect: The run is 14.4 ± 0.1 m, so the true run lies between 14.3 and 14.5 m. Either way it exceeds 9 m, so the violation (if no landing exists) is robust to measurement error - the uncertainty band does not straddle the threshold. This is the key distinction from the angle check: there the band straddled the limit, here it does not.
Inspector's judgment: The angle is within tolerance but barely; the landing rule is a clear pass or clear fail depending on physical layout. The inspector should document the angle as marginal, verify landings by direct observation, and note that rounding conventions (truncating 4.76° to 4.7° versus rounding to 4.8°) could change a borderline verdict - so the raw ratio, not the rounded angle, should govern.