The Ramp Angle Dispute
A wheelchair ramp is being designed for a community centre entrance. The horizontal run is 6.0 m and the rise is 0.9 m, giving an incline angle θ with tan θ = 0.15. A builder proposes increasing the run to 8.0 m while keeping the rise, claiming "the angle halves because the run increases by a third." A code consultant says the maximum permitted angle is 8.5°. The architect also wants a handrail whose length equals the slope length, and a landing at the top where the ramp meets a 1.2 m high platform. A surveyor measures the existing ramp angle as 8.53° and the consultant flags it as borderline. The team must decide whether the proposed 8.0 m run satisfies the code and how to report the angle precisely.
1. Compute the angle for both the 6.0 m and 8.0 m runs, and evaluate the builder's claim that the angle halves.
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Original: tan θ = 0.9/6.0 = 0.15, so θ = arctan(0.15) ≈ 8.531°.
Proposed: tan θ = 0.9/8.0 = 0.1125, so θ = arctan(0.1125) ≈ 6.419°.
The angle does not halve: 8.531/2 = 4.27°, far below 6.42°. The tangent is not linear, so proportional changes in run do not produce proportional changes in angle. The builder's reasoning assumes linearity where the relationship is nonlinear. The judgement is to recognize that small-angle approximations (tan θ ≈ θ) can mislead when comparing ratios; here the angle changed by about 25%, not 50%.
2. Determine whether each design meets the 8.5° maximum, and compute the handrail length for the 8.0 m run.
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The 6.0 m run gives 8.531°, which exceeds 8.5° by about 0.03°, so it fails the code by a hair. The 8.0 m run gives 6.419°, comfortably within 8.5°.
Handrail (slope length) for the 8.0 m run: L = √(8.0² + 0.9²) = √(64 + 0.81) = √64.81 ≈ 8.050 m.
For the original: L = √(36 + 0.81) = √36.81 ≈ 6.067 m.
The judgement call is that the surveyor's 8.53° and the computed 8.531° agree, so the original ramp is non-compliant, and the borderline nature means rounding could mask the violation. Reporting to two decimal places shows the breach; rounding to the nearest degree (9°) would also show it, but rounding to 8.5° would hide it. Precision matters at a code boundary.