Parabolic Tunnel Cross-Section and Clearance for a Light Strip
A pedestrian tunnel has a cross-section modeled by a parabola. The floor is horizontal, the tunnel is 12 m wide at floor level, and the highest point is 8 m above the floor at the center. A maintenance team wants to install a thin horizontal light strip whose center is on the tunnel's vertical centerline. The strip may not touch the curved roof, and a safety rule requires at least 0.4 m of vertical clearance between the roof and each end of the strip. The team first tests a 6 m wide strip at height 6 m and finds that its ends just touch the roof, so that height is unacceptable. A supplier offers a 7 m wide strip, and the team also needs the maximum height for a 5 m wide strip. The team must build a coordinate model, use it to answer both questions, and explain why a linear approximation near the crown would give an unsafe answer. All dimensions are in meters, and the tunnel is symmetric about its vertical centerline.
1. Choose coordinates with the floor at y = 0 and the centerline at x = 0. Find the equation of the parabolic roof and the maximum height h for a 7 m wide strip that satisfies the 0.4 m clearance rule.
Read the worked answer
Place the floor on y = 0 and the centerline on x = 0. The tunnel meets the floor at x = −6 and x = 6, and the vertex is (0, 8). A downward-opening parabola with vertex at (0, 8) has the form y = 8 − kx^2.
Using the floor point (6, 0): 0 = 8 − k(36), so k = 8/36 = 2/9. Thus y = 8 − (2/9)x^2.
For a 7 m wide strip centered at x = 0, the ends are at x = ±3.5. The roof height there is y = 8 − (2/9)(3.5)^2 = 8 − (2/9)(12.25) = 8 − 24.5/9 = 5.277... m.
The strip height h must leave at least 0.4 m below the roof at its ends, so h ≤ 5.277... − 0.4 = 4.877... m. Therefore the maximum height is about 4.88 m.
2. Find the maximum height for a 5 m wide strip under the same clearance rule.
Read the worked answer
For a 5 m wide strip, the ends are at x = ±2.5. The roof height there is y = 8 − (2/9)(2.5)^2 = 8 − (2/9)(6.25) = 8 − 12.5/9 = 6.611... m.
Applying the 0.4 m clearance: h ≤ 6.611... − 0.4 = 6.211... m. Thus the maximum height is about 6.21 m. This is higher than for the 7 m strip because the ends are closer to the crown, where the roof is higher.
3. Explain why replacing the parabola near the crown by its tangent line would give an unsafe clearance estimate.
Read the worked answer
At the crown x = 0, the parabola has its maximum, so the tangent is horizontal: y = 8. Near the crown the tangent line predicts that the roof is still almost 8 m high even at x = 3.5 or x = 2.5, whereas the actual parabola has already dropped.
For example, at x = 3.5 the tangent predicts y = 8, but the true roof height is about 5.28 m. A planner using the tangent might allow a strip at h = 6 m and expect about 2 m of clearance, while the actual clearance at the ends would be 5.28 − 6 < 0, meaning the strip would intersect the roof. The tangent is safe only extremely close to the center; away from the vertex the curvature must be included.