Fitting a Parabola to a Cable Sag
A suspension bridge has a main cable that, between two identical towers, is modelled by a parabola. The towers are 120 m apart and each is 30 m tall above the deck. At the midpoint between the towers the cable dips to 6 m above the deck. An engineer proposes instead a second model, a parabola with the same towers and midpoint height but whose axis of symmetry is shifted 10 m to the right of the midpoint, arguing this better reflects an asymmetric load. A student must decide which model is physically sensible, and use it to find the clearance above the deck at a point 20 m horizontally from the left tower.
1. Explain why only one of the two proposed parabolic models is consistent with the described cable, and state the equation of the valid model.
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A freely hanging symmetric cable between equal-height towers must be symmetric about the midpoint, so its axis of symmetry is the vertical line through the midpoint. Model 1 has vertex at the midpoint and equal tower heights, which is consistent. Model 2 keeps the same three key heights (left tower 30 m, midpoint 6 m, right tower 30 m) but places its axis 10 m to the right, which would force the cable to be higher on one side than the other between towers; it cannot pass through both 30 m tower tops and the 6 m midpoint while being shifted. So Model 1 is valid.
Set origin at the left tower base on the deck, x in metres. Vertex at x = 60, y = 6: y = a(x − 60)^2 + 6. Use (0, 30): 30 = a(3600) + 6, so a = 24/3600 = 1/150. Thus y = (1/150)(x − 60)^2 + 6.
2. Using the valid model, find the clearance 20 m from the left tower, and interpret whether a 12 m tall truck could pass at that point.
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Substitute x = 20: y = (1/150)(20 − 60)^2 + 6 = (1/150)(1600) + 6 = 10.667 + 6 ≈ 16.67 m. So clearance ≈ 16.7 m. A 12 m truck needs clearance strictly greater than 12 m plus safety margin; 16.7 m exceeds 12 m, so it could pass with roughly 4.7 m spare. The judgement call is recognising that the truck height is compared to cable height, not deck-to-tower height, and that the vertex value (6 m) is the minimum, so near midspan a 12 m truck would NOT fit (6 m < 12 m). Clearance is not uniform.