Math CentralQuandaries & Queries


Question from Jessica, a student:

I have  trouble solving this problem, could you help me. This is College Algebra .

A suspesion bridge with weight uniformly distributed along its length has twin towers that extend 75 meters abouve the road surfce and are 400 meters apart.The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 100 meters from the center. (Assume that the road is level.)

Formula is : y=ax^2
the answer is: 18.75 meters

Hi Jessica.

Draw a parabola on graph paper where the center of the bridge is at the origin and the cables rise up to to the points (±200, 75). Thus, the x axis is the road and the height of the parabola above the x axis (this would be the y value) reflects the height of the cable above the road surface.

You know that the equation of a parabola that opens up is in the form y - yV = a(x-xV)2, where (xV, yV) is the vertex (in this case, the origin) and a is just a constant that determines the curvature.

Thus, the equation is y = ax2. You need to find out what a is.

To do this, you need to put the coordinates of a known point on the parabola (other than the vertex) into the equation in place of x and y and then solve for a.

When you have the proper equation, you can substitute the value x = 100 and find the corresponding height of the cables above the road surface 100 m from the center of the road (the origin). If you do the arithmetic properly, you will indeed get 18.75 m.

Stephen La Rocque.

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