Bridge Cable Parabola Inspection
A suspension bridge has a main cable modelled by the parabola y = 0.004x² - 0.48x + 20, where x is the horizontal distance in metres from the left tower and y is the cable height above the deck in metres. The cable is anchored at the left tower (x = 0) and the right tower (x = 120). An inspector measures the lowest point of the cable and finds it is 5.6 m above the deck. The design code requires the cable's lowest point to be at least 6 m above the deck for clearance. The bridge company claims the cable satisfies the code because the endpoints are 20 m high. A second concern is whether the cable's lowest point is horizontally centred between the towers.
1. Find the coordinates of the vertex of the parabola and determine whether the cable meets the 6 m clearance code.
Read the worked answer
For y = ax² + bx + c with a = 0.004, b = -0.48, c = 20, the vertex x-coordinate is x = -b/(2a) = 0.48/(0.008) = 60.
Substitute x = 60: y = 0.004(3600) - 0.48(60) + 20 = 14.4 - 28.8 + 20 = 5.6.
So the vertex is (60, 5.6), matching the inspector's measurement. The lowest point is 5.6 m above the deck, which is below the required 6 m.
The company's claim is wrong: high endpoints do not guarantee adequate mid-span clearance. The cable fails the code by 0.4 m. The lowest point is at x = 60, exactly halfway between the towers at x = 0 and x = 120, so the cable is horizontally centred.
2. The company proposes raising the whole cable by 0.5 m. Determine the new equation, check the code, and comment on whether this changes the position of the lowest point.
Read the worked answer
Raising the cable by 0.5 m adds 0.5 to every y-value: y = 0.004x² - 0.48x + 20.5.
The vertex x-coordinate is unchanged because a and b are unchanged: x = 0.48/0.008 = 60. The new vertex height is 5.6 + 0.5 = 6.1 m.
Since 6.1 ≥ 6, the cable now satisfies the clearance code, with 0.1 m to spare.
Judgement: the fix works and keeps the lowest point centred at x = 60. However, the margin is only 0.1 m, which leaves little room for construction tolerance, cable sag under load, or thermal expansion. A more robust design would raise it further or reduce the sag. The proposal is technically valid but marginal.