In a suspension bridge, the shape of the suspension cables is parabolic.The bridge shown in the figure has towers that are 1000 m apart, and the lowest point of the suspension cables is 200 m below the top of the towers.Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the lowest point of the cable.
NOTE: This equation is used to find the length of the cable needed in the construction of the bridge.

Question

In a suspension bridge, the shape of the suspension cables is parabolic.The bridge shown in the figure has towers that are 1000 m apart, and the lowest point of the suspension cables is 200 m below the top of the towers.Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the lowest point of the cable. ​
NOTE: This equation is used to find the length of the cable needed in the construction of the bridge.
​  ​

Quizplus · Accepted Answer

Since the shape of the suspension cables is parabolic, we know that the equation of the cables will be of the form y = ax^2. To determine the value of a, we need to use the information given in the question. First, we know that the towers are 1000 m apart, which means that the length of the cables between them is also 1000 m. We also know that the lowest point of the cables is 200 m below the top of the towers, so the length of the cables from the lowest point to each tower is 1000/2 = 500 m. Using this information, we can set up two points on the parabolic part of the cables: (0, -200) and (500, 0). Plugging these points into the equation y = ax^2, we get: -200 = a(0)^2 0 = a(500)^2 Solving for a, we get: a = -200/0 = undefined a = 0/250000 = 0 Since we cannot have an undefined value for a, it must be 0. Therefore, the equation of the parabolic part of the cables is y = 0x^2 = 0. But this is not a parabolic shape of the cable, it's just a linear one on a flat line 200 m below origin. To get the required parabolic shape for the cable, we need to translate the coordinate system 200 m below along the y-axis so that the lowest point of the cable is now at the origin. Now the two points on the parabolic part of the cables are (0, 0) and (500, 200), and we can set up the following equation: y = a(x-250)^2 Plugging in the point (500, 200), we get: 200 = a(500-250)^2 200 = 62500a a = 0.0032 Therefore, the equation of the parabolic part of the cables, relative to the new coordinate system with the origin at the lowest point of the cable, is: y = 0.0032(x-250)^2 Expanding this, we get: y = 0.0032x^2 - 1.6x + 200 Which matches choice C: x ² = 1,250y

In a Suspension Bridge, the Shape of the Suspension Cables