Question

Write the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.

Short answer

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

Step-by-step solution

Step 1. Given Information

Given parabolic arch formed in the founded of the bridge is

Step 2. Constructing the parabola

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 $\left(50,50\right)$.

Step 3. Equation of the parabola

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:

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

Step 4. Conclusion

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