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. A)x² = -1,250y B)x² = 1,250y C)x² = 5,000y D)x² = 2,500y E)y² = 5,000x

Question

Quizplus · Accepted Answer

The shape of the suspension cables is parabolic, and we are told that the lowest point of the cable is 200 m below the top of the towers. This means that the vertex of the parabola is located 200 meters below the top of the towers, which is also the origin of our coordinate system.

We are also given that the towers are 1000 meters apart, so the x-coordinate of the two points where the parabolic cables connect to the towers will be -500 and 500.

Let's call the y-coordinate of the vertex "h". Since the parabola is symmetric around its vertex, the y-coordinate of the two points where the parabolic cables connect to the towers will also be "h".

Using the standard form equation for a parabola, y = a(x-h)^2 + k, we know that the vertex is (0, -200) which gives us:

-200 = a(0 - 0)^2 + h

h = -200

Now we need to find the value of "a". To do this, we can use one of the points where the parabolic cables connect to a tower. Let's use the point (-500, -200) since it has the smallest absolute value for the x-coordinate.

-200 = a(-500 - 0)^2 - 200

-200 = 250000a

a = -0.0008

Therefore, our equation for the parabolic part of the cables is:

y = -0.0008(x^2) - 200

or

x^2 = -1250y

In a Suspension Bridge, the Shape of the Suspension Cables