Arch Culvert Clearance: Does the Rice Truck Fit?
The provincial road office in Kampong Cham is repainting clearance markings on a 20 m span concrete arch culvert on National Road 7. An engineer models the underside of the arch, in metres above the road surface, as h(x) = 4 − 0.04x², where x is the horizontal distance in metres from the centreline; the arch meets the road surface at x = ±10 m. A rice truck 2.50 m wide and 2.80 m high, including its load, is driven through with its centreline 5.00 m to the right of the road centreline. The driver argues that the truck fits, because the arch is exactly 3.00 m high directly above his centreline, which is more than the 2.80 m he needs. The office also wants a general rule giving the largest sideways offset allowed for any truck of this size, keeping a 0.20 m safety margin below the arch. Recent resurfacing has raised the road surface by 0.05 m at the crown, and this has not yet been subtracted from the model.
1. Over what horizontal range is the clearance at least 2.80 m? Set up and solve the inequality, giving exact and rounded values, and state the width of the usable band.
Read the worked answer
Clearance is h(x) = 4 − 0.04x², and we need h(x) ≥ 2.80:
4 − 0.04x² ≥ 2.80 0.04x² ≤ 1.20 x² ≤ 30 |x| ≤ √30 = 5.477 m
So the band is 5.48 m to each side of the centreline, about 10.95 m wide in total, out of a 20 m span. Two checks matter here. First, the endpoints are included: at x = √30 the clearance is exactly 2.80 m, which leaves zero margin, so a truck of exactly this height should not be driven at that offset. Second, this band describes the roof line of the truck, not its centre. A driver who reads 5.48 m as 'my centre may be 5.48 m off-line' makes a category error that the next task exposes.
2. Test the driver's claim at a 5.00 m offset. Then derive the maximum permitted centre offset (a) with no margin and (b) with the 0.20 m safety margin, and state how the 0.05 m resurfacing changes each answer.
Read the worked answer
At a 5.00 m centre offset the truck's outer edge is at x = 5.00 + 1.25 = 6.25 m, and its inner edge at 3.75 m.
h(6.25) = 4 − 0.04(39.0625) = 4 − 1.5625 = 2.4375 m h(3.75) = 4 − 0.04(14.0625) = 3.4375 m
The inner edge clears, but the outer corner has only 2.44 m against a 2.80 m truck: the roof would strike the arch. The driver checked only the single point above his centreline, so his reasoning is invalid.
General rule, worst corner at c + 1.25: (a) No margin: (c + 1.25)² ≤ 30, so c ≤ 4.227 m, rounded down to 4.22 m. (b) Safety margin: need clearance ≥ 3.00 m, so (c + 1.25)² ≤ (4 − 3.00)/0.04 = 25, giving c ≤ 3.75 m.
Resurfacing: height becomes h'(x) = 3.95 − 0.04x². Then (a) x² ≤ 28.75, x ≤ 5.362, c ≤ 4.11 m; and (b) x² ≤ 23.75, x ≤ 4.873, c ≤ 3.62 m. Always round offsets down, never up.
3. A colleague says the parabolic model can be trusted up to the abutments at x = ±10 m. Give one mathematical and one physical reason for treating the model with caution there, and say what this implies for the painted rule.
Read the worked answer
Mathematically, the parabola meets y = 0 with slope dh/dx = −0.08x, which at x = 10 m is −0.80. That is an angle of about 39° below horizontal, so the model says the arch leans in gently and the last half-metre of headroom is won slowly. Real culverts spring from near-vertical abutment walls, so the true clearance close to the wall is greater than the parabola predicts - the model is conservative there, but it is also simply the wrong shape, and any marking painted on the assumption of a smooth curve will mislead drivers about how quickly headroom returns.
Physically, the model ignores things that matter at the edges: the road camber tilts the truck, the load may be stacked unevenly or shift, and the driver's path wanders by tens of centimetres. A single painted number derived from a smooth centreline path therefore overstates safety.
Practical implication: publish the rule from part (b) only for the central portion of the span - roughly |x| ≤ 8 m - and mark the outer 2 m as a no-overtaking, low-clearance zone rather than quoting a parabolic offset that drivers will treat as a target.