Question

The towers of a suspension bridge are 450 feet apart and 150 feet high from the roadway. Cables are at a height of 25 feet above the roadway, midway between the towers, but gradually get taller toward each end. Assume the x-axis is the roadway and the y-axis is the center of the bridge, write an equation for the parabola. What is the height of the cable at a point 50 feet from one of the towers? Round to the nearest whole number. 

Answer 1

The question implies that the shape is a parabola, not a catenary, which is the natural shape of a non-rigid material suspended between towers.

With the y-axis at the centre the equation of the parabola is of the form y=ax²+b, where an and b are positive constants. The vertex is on the axis of symmetry, which is the y-axis and can be represented by the point (0,b) because when x=0 (the midpoint of the bridge) y=b, the minimum height above the roadway. 

(0,b)=(0,25) so b=25ft, and the equation becomes y=ax²+25. Now we need to find a.

The towers are 450ft apart and are each 150ft high above the road. If we assume that the parabola is formed from the top of each tower, then the points (-225,150) and (225,150) are graphical representations of points on the parabola so we can plug in the values (x=±225, x²=225²=50625).

150=50625a+25, 50625a=125, a=125/50625=1/405.

So y=x²/405+25 is the equation of the parabola.

a point 50ft from either tower is an x-coord of 225-50=175ft.

y=175²/405+25=8150/81ft. This corresponds to the point on the parabola (175,8150/81). This is the height of the parabola above the road, which is also the length of the cable hanging from the parabola=100.62ft or 101ft to the nearest foot.