Semi-Elliptical Tunnel: Truck Clearance at the Upper Corners
A road tunnel has a semi-elliptical cross-section. The width at road level is 12 m, and the maximum height at the centre is 4 m. The ellipse is centred on the road, with the road as the major axis. A transport company wants to send a rectangular truck 3 m wide and 3.8 m high through the tunnel. Because of lane barriers the truck must stay centred, so its upper corners are at x = ±1.5 m, where x is measured from the centre. A second truck is 5 m wide and 3.6 m high. The tunnel authority requires at least 0.25 m vertical clearance above the truck at its widest point. The company argues that because the tunnel is 4 m high at the centre, any truck under 4 m must fit. The authority wants a calculation, not a guess. Assume the truck body is rectangular and the road is flat.
1. Derive the equation of the elliptical cross-section and find the tunnel height at x = 1.5 m.
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Take the centre at the origin and the road as the x-axis. The semi-ellipse has semi-major axis a = 12/2 = 6 m along x and semi-minor axis b = 4 m along y, so its equation is
x²/36 + y²/16 = 1, y ≥ 0.
Solving for y:
y = 4 sqrt(1 − x²/36).
At the truck's widest point, x = 3/2 = 1.5 m, so
y = 4 sqrt(1 − 2.25/36) = 4 sqrt(1 − 1/16) = 4 sqrt(15/16) = sqrt(15) ≈ 3.873 m.
So the tunnel is about 3.873 m high at x = 1.5 m, not 4 m.
2. Check both trucks against the 0.25 m clearance rule. Which, if any, is allowed?
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First truck: width 3 m gives x = 1.5 m, where the tunnel height is sqrt(15) ≈ 3.873 m. Clearance = 3.873 − 3.8 = 0.073 m, which is less than 0.25 m. It is not allowed under the rule, although its upper corners are just inside the ellipse.
Second truck: width 5 m gives x = 2.5 m. Then
y = 4 sqrt(1 − 6.25/36) = 4 sqrt(1 − 25/144) = 4 sqrt(119/144) = (4/12) sqrt(119) = sqrt(119)/3 ≈ 3.636 m.
Clearance = 3.636 − 3.6 = 0.036 m, also less than 0.25 m. This truck is not allowed either. The company's argument fails because the critical height is at the truck's upper corners, not at the tunnel centre.
3. Find the maximum height of a 3 m wide truck that satisfies the clearance rule.
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For a 3 m wide centred truck, the critical point is x = 1.5 m, where the tunnel height is sqrt(15) m. To leave 0.25 m clearance, the maximum truck height is
sqrt(15) − 0.25 ≈ 3.873 − 0.25 = 3.623 m.
So a 3 m wide truck must be no taller than about 3.62 m to satisfy the rule. A truck taller than that violates the clearance requirement even though it may still be shorter than the 4 m centre height.