In a suspension bridge, the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 400 m apart, and the lowest point of the suspension cables is 100 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. A)x² = 800y B)x² = 400y C)x² = -400y D)x² = 1,600y E)y² = 1,600x

Question

Quizplus · Accepted Answer

Since the towers are 400m apart, the distance from the origin to each tower is 200m. Also, since the lowest point of the cable is 100m below the top of the towers, the vertex of the parabola is at (0,-100). Therefore, the equation of the parabolic part of the cable can be written in the form:

y = a(x-0)^2 - 100

We know that the parabola passes through the two towers, which are located at (-200,0) and (200,0). Plugging these points into the equation gives us:

0 = a(-200-0)^2 - 100 
0 = a(200-0)^2 - 100

Simplifying these equations gives us:

a = 1/800

Substituting this value of a into the equation y = a(x-0)^2 - 100 and simplifying gives us:

y = (1/800)x^2 - 100

Multiplying both sides by 800 gives the final equation:

x^2 = 400y, which can be written as 2S1S1P0 = 400y, where S1 is the midpoint of the line segment connecting the two towers. Therefore, the correct choice is B.

In a Suspension Bridge, the Shape of the Suspension Cables