The prompt is to find the parabola equation for the main span of Golden Gate bridge. Based on the $3$ points on the parabola: $(0,227)$; vertex $(640,75)$; and $(1280,227)$ I found the equation:
$$y = 0.00037109375x^2-0.475x+227$$
It is required to calculate the heights of vertical ropes under the parabolic main cable, which spans across a length of $1280$ m.
Vertical ropes are distanced $15.24$ m apart with a diameter of $0.09736667$ m. The $x$-range is $[0,1280]$.
If I assume that the heights of vertical ropes follow the geometrical progression I get that $u = 219.8471693*0.968^{n-1}$. Where $u=$ height of vertical rope, $n=$ the number of the vertical rope in sequence.
I need a way to to prove if this equation is correct or not.
I tried by writing that \begin{align}219.8471693\cdot 0.968^{n-1} = & 0.00037109375\cdot(15.24n+0.09736667\cdot (n-1))^2 \\ & -0.475 \cdot (15.24n+0.09736667\cdot(n-1))+227\end{align} where L.H.S. is the function of height of vertical rope calculated by geometrical induction and R.H.S. is the height of vertical ropes calculated by using parabola function. $x$ at this point is the position of vertical rope on $x$-axis which is calculated by using the distance between ropes and diameter of one rope, and $n$ is the number of rope in the sequence. I tried to prove this equality by mathematical induction, but I couldn't.
How could it be proven that the heights of vertical ropes follow or not the geometrical progression?
Ignoring (x,y) shift
$$ y= k x^2, \quad k= y/x^2 $$
$$ k = (227. - 75)/640^2 = 0.000371094 $$
Considering $(x,y)$ shift $ = (h,k)$
$$ (y - k) = k (x - h)^2 $$
$$ (y - 75) = k (x - 640)^2 $$
The above is a parabolic progression. Please check your calculation. If a table of values (x,y) is given a parabola can be verified by numerically taking first differences twice ( like differentiating twice) from a table of (x,y) values.
Geometric progression is for an exponential curve!
We have a deep catenary suspension bridge cable of shape $ y =c \cosh (x/c)$ if rate of loading $ dQ/ds = $ constant with respect to sloping arc/arch/cable direction and is of shallow parabolic cable shape $ y= c x^2 $ if $ dQ/dx =$ constant for constant loading with respect to span , the x direction. For the Golden Gate bridge the deck is horizontal and bears a constant load per meter, the error is not so high for its shallow arch and the latter is more accurately described as a parabolic bridge.