The Suspension Bridge Cable
An engineer models the main cable of a suspension bridge. The two towers are 80 m apart and 30 m tall, and the cable is attached at the top of each tower. Between the towers the cable hangs in the shape of a parabola with its lowest point 10 m above the deck, which runs level between the tower bases. The engineer first writes the cable height as a quadratic in the form y = a(x − h)² + k, using the midpoint between the towers as the origin of x. A student assistant instead claims the cable can be modelled by a piecewise function made of two straight segments meeting at the lowest point, arguing that 'parabolas are only for projectile motion'. The engineer must decide which model to use before computing clearance for traffic, and also needs to know at what horizontal distance from the centre the clearance equals 20 m.
1. Determine the quadratic model and state why the piecewise-linear model is unsuitable for the clearance calculation.
Read the worked answer
Place the origin at the midpoint, so the towers sit at x = ±40. The vertex is (0, 10), giving y = a x² + 10. Using the tower top (40, 30): 30 = 1600a + 10, so a = 20/1600 = 1/80. Hence y = x²/80 + 10.
The piecewise-linear model is unsuitable because a real hanging cable is smooth at its lowest point; two straight segments create a corner (a cusp in the gradient) at x = 0, and the gradient of the cable there must be zero by symmetry. The parabola has dy/dx = x/40, which vanishes at x = 0 as required. A linear model would also underestimate clearance away from the centre in a way that depends on where you switch segments, making the answer arbitrary. The quadratic is the correct choice.
2. Find the horizontal distance from the centre at which the cable clearance is exactly 20 m, and comment on the physical meaning of both solutions.
Read the worked answer
Set y = 20: x²/80 + 10 = 20, so x²/80 = 10, x² = 800, x = ±20√2 ≈ ±28.28 m.
Both solutions are physically meaningful: the cable reaches 20 m clearance at 28.28 m on either side of the centre. Since the towers are at ±40 m and the cable is symmetric, the 20 m clearance occurs twice, once on the rising part of the cable on each side. The clearance is at least 20 m only for |x| ≥ 20√2 within the span; near the centre the clearance drops to 10 m. This matters for traffic: the lowest clearance governs the height limit for vehicles, and the two solutions confirm the symmetry of the model rather than indicating an error.