Tricky question about calculating height in parabolas: 10 POINTS!?

Question

a design for a bridge over a river consists of a steel arch in the shape of a parabola . If the base of one of the pylons is taken as the origin, the height (h) of the arch is given by h= -0.01y²+ 2.7 where "y" is the horizontal distance across the river. The road is to be situated 50m above the base of the pylon. There are 22 vertical steel cables equally spaced between the road and the arch , in the parabola.

Calculate the length of steel cable required for these supports.

Answer 1

Well, first you have to calculate the value of y at the positions of each of the cables.  You can calculate the total span because h = 0 at the far end also, so divide that by 21  (if there are 22 cables then there must be 21 gaps between them)  to get the spacings between the cables.

Then, at each position, work out the length of the cable, and take the total.  (Or actually, since it's symmetrical, you need only do the first eleven cables and double it:  the lengths of the first and 22nd, second and 21st, third and 20th, 4th and 19th and so forth will be equal.)

Answer 2

Assuming the height of the bridge is 2.7m at the center and there are 11 cables on each side of the center line. First:

  h = 2.7-0.01*y^2 would equal zero at y^2 = 270 or y = +/- 16.43m leaving a dy of 16.43/12 (there are 11 lines, but 12 segments to complete y). So, the calculations (very easy in Excel) are

h = 2.7 - (0.01)[(line#)(16.43/12)]^2. Perform this calculation for 1 to 11 and sum the results. Double the sum to count both sides of the bridge. I get 40.425m of cable.

Answer 3

The problem should be straight forward to solve, but I'm not sure I understand the bridge design correctly, please clarify.

I'm imagining a bridge like this one: http://farm2.static.flickr.com/1224/1350...

According to your formula for the arch, the peak of the arch is 2.7m above the water (where y=0), right? You say there is a pylon at the origin. Why would an arch bridge have a pylon right in the middle of the arch? If the road is 50m above the water, it would be above the arch, which doesn't make sense.

If you can describe the bridge a bit more or add a link to an image of the bridge you're describing, I can help to answer the question.