SITUATIONAL PROBLEM INVOLVING PARABOLA

Question

The cables of the suspension bridge are in the shape of parabola. The towers supporting the cable are 100 ft. apart and 100 ft. long. If the cables are at height of 10 ft. midway between the towers, what is the height of the cable at a point 50 ft. from the center of the bridge? What is the standard equation of the parabola that models a cable?

Answer 1

I find the best way to begin problems like this is to draw them on a piece of graph paper.

We can begin by centering the two buildings on the origin (0,0), so one building would be at (50,0) and the other at (-50,0). We know that since the cables are likely attached to the tops of the buildings the parabola they create must hit the points (50,100) and (-50,100). And we're also given the point (0,10) since at the midway — the origin — the parabola is 10 feet up.

We can use the vertex form of a parabolic equation to write the equation:

y=a(x−h)²+k

The point (h,k) is the vertex or the minimum/maximum point of the parabola. In this case, we know that the minimum point is (0,10).

y=a(x-0)²+10

y=ax²+10

Since we know another point that the parabola passes through (50,100) we can input those numbers for x and y and do some algebra to find the value of a.

100 = a(50)²+10

100-10 = a(50)²

90 = a(2500)

90/2500 = a

9/250 = 0.036 = a

So our final equation should look like...

y=0.036x²+10

We can check our equation by graphing it over our previous picture. Does it fit? Good! Now we can use our new equation to solve for the height (y) of the bridge at any distance from the midpoint (x).