I am solving the following word problem:
"A high voltage cable is supported by two towers $2800$ feet apart and $348$ feet high. The cable hangs in approximately the shape of a parabola, and the lowest point on the cable is $200$ feet above the ground. Find the equation of the parabola."
So I didn't have luck solving it myself. The answer says to use the equation $y=\frac{x^2}{4p}$ and setting $x$ to $1400$ and $y$ to $148$. I don't understand why you set $y$ to $148$ instead of $200$. I would have thought that it would make more sense to plot the vertex of the parabola at $y=200$ with the origin of the graph at ground level. What would the reason for plotting at $148$ be?
Secondly, the correct answer appears to be $y=37x^2/490000$. However when I solve $y=x^2/4p$ for $p$ using the values of $y=148$ and $x=1400$, I end up with $p=490000/37$, which would lead me to think that I should multiply $p$ by 4 when substituting back into the parabolic equation and the correct answer should be $y=37x^2/1960000$. Is this approach incorrect?
According to the figure I drew below you can see that since the parabola is symmetric about the y-axis and there is 2800 between the legs then it would be 1400 feet the distance between the end of the cable and the x-axis and -1400 from the other side. The answer says set $y$ to $148$ because they want the parabola to touch the x-axis, i.e, the lowest point of the parabola would be $(0,0)$. If I want to solve it I will consider the function $f(x) = ax^2 + b$ and find $a,b$ from the points $(0,200), (1400, 348)$. But in your case they consider the function $f(x) = ax^2$ and the points $(0,0), (1400,148)$.