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masters Draw the parabola with the y-axis as axis of symmetry and the bed of the road will be the x-axis. See diagram.
The points (-50, 0) and (50, 0) line on the parabola on the x-axis since the span is 100.
The points (-40, 10) and (40, 10) also lie on the parabola.
We use $\displaystyle (x-h)^2=4p(y-k)$ for our equation of a parabola with vertex (h, k) since the axis if symmetry is vertical. We know our vertex is at (0, k). We need to find k.
Substituting point (50, 0) into this equation, we get:
$\displaystyle (50-0)^2=4p(0-k)$
$\displaystyle \boxed{2500=0p-4pk}$
Substituting point (40, 10) into this equation, we get:
$\displaystyle (40-0)^2=4p(10-k)$
$\displaystyle \boxed{1600=40p-4pk}$
Use the two boxed equations to solve for p.
$\displaystyle 2500= \ \ 0p-4pk$
$\displaystyle 1500=40p-4pk$
Subtract the two equations to get:
$\displaystyle 900=-40p$
$\displaystyle p=-\frac{45}{2}$
Now, to find k, we substitute p back into one of our boxed equations.
$\displaystyle 2500=0\left(\frac{45}{2}\right)-4\left(-\frac{45}{2}\right)k$
$\displaystyle 2500=90k$
$\displaystyle k=\frac{350}{9} \approx 27.8$ feet.