Hard math question need help?

The generic formula of a parabola is y=x^2. In this case this is an inverted parabola with known “end points”, so if this is a true parabola, then the formula would look like:

y = h – kx^2

Where h is the max height of the road and k is the parabolic constant in question.

I’m going to set up my origin so that it is in the middle of the road. So that when x = 0 is where we will find our maximum height h.

Because of the cable, you are given a point on the parabola that exists where y = 9.4cm and x = (-760cm/2) + 156cm = -224cm.*

* Why? Because we are assuming that the max height is in the middle of the road at the hump and we know that the road is 7.6m across (7.6m = 760cm), then our “0” point is in the middle at 760cm / 2 = 380cm. So in this co-ordinate system, our left edge is at -380cm, so the cable being 156cm from the edge lives at -224cm.

ok:

x1 = -224cm, y1 = 9.4cm

x2 = -380cm, y2 = 0cm

Both of these points fit on the parabola described by:

x1,y1 (-224cm, 9.4cm)

y = h – k(x^2)

9.4 = h – k(-224^2)

9.4 = h – 50176k

h = 9.4 + 50176k

x2,y2 (-380cm,0cm)

0 = h – k(-380^2)

0 = h – 144400k

h = 144400k

So now we have 2 equations and 2 unknowns.

h = 9.4 + 50176k

h = 144400k

Solving for k.

9.4 + 50176k = 144400k

9.4 = 94224k

k = .000099762293

Plugging this value into the formula h = 144400k

h = 14.4056752536cm which is by definition of our parabola’s origins, should be the max height of the parabola.

Please double check the math, but the idea I believe is identifying the generic form of a parabola and then realizing that if you select your origin in the right place that you can identify at least two points that make sense. In this case, the points are where the cable is embedded where you know the height, and the approximation that the edge of the road is at ground zero. Plugging these values into your parabolic equation gives you a 2 equations and 2 unknowns situation where you can solve for the height h and the parabolic constant k.

Hope this helps or if someone can pick up the ball and simplify this explanation.

Good luck!

Huh