91Ó°ÊÓ

Given parabolic arch formed in the founded of the bridge is

Consider the left lower end of the bridge to be at the origin i.e. $\left(0,0\right)$.

Let the base be along the x axis. Since the base of the bridge is 100 feet wide, the other end of the bridge is at $\left(100,0\right)$

The highest point of the bridge is the vertex and is 50 feet high. So the y coordinate of the vertex is 50. The vertex lies midway between the end points on the base i.e. 50 feet from the origin.

Hence the vertex is $\left(50,50\right)$.

The standard equation of a parabola is $y=a{\left(x-h\right)}^2+k$

Where the vertex is $\left(h,k\right)$.

Plugging the values of vertex point in the equation:

$$\begin{gathered} y=a{\left(x-h\right)}^2+k \\ y=a{\left(x-50\right)}^2+50 \end{gathered}$$

The parabola passes through $\left(0,0\right)$.

Plugging the values in the equation to determine the value of a:

$$\begin{gathered} y=a{\left(x-50\right)}^2+50 \\ 0=a{\left(0-50\right)}^2+50 \\ a\left(2500\right)=-50 \\ a=-\frac{50}{2500} \\ a=-\frac{1}{50} \end{gathered}$$

The equation of the parabola in standard form is$y=-\frac{1}{50}{\left(x-50\right)}^2+50$