91Ó°ÊÓ

Consider the given question,

Maximum height of the arch is $25ft$.

The vertex form of the equation of a parabola facing downwards is $f\left(x\right)=-a{\left(x-h\right)}^2+k......\left(i\right)$

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

Draw the figure,

Substitute $\left(0,25\right)$in equation (i),

role="math" localid="1647370239069" $$\begin{gathered} f\left(x\right)=-a{\left(x-0\right)}^2+25 \\ f\left(x\right)=-a{x}^2+25......\left(ii\right) \end{gathered}$$

Substitute $f\left(60\right)=0$in equation (ii),

role="math" localid="1647370288858" $$\begin{gathered} 0=-a{\left(60\right)}^2+25 \\ a=\frac{25}{3600} \end{gathered}$$

Substitute the value of a in equation (ii),

role="math" localid="1647370435546" $f\left(x\right)=-\frac{25}{3600}{x}^2+25......\left(iii\right)$

Thus, the equation of the parabolic arch is$f\left(x\right)=-\frac{25}{3600}{x}^2+25$.

Substitute $x=10$in equation (iii),

role="math" localid="1647370481192" $$\begin{gathered} f\left(10\right)=-\frac{25}{3600}{\left(10\right)}^2+25 \\ f\left(10\right)=24.31ft \end{gathered}$$

Substitute $x=20$in equation (iii),

role="math" localid="1647370518729" $$\begin{gathered} f\left(20\right)=-\frac{25}{3600}{\left(20\right)}^2+25 \\ f\left(20\right)=22.22ft \end{gathered}$$

Substitute $x=30$in equation (iii),

$$\begin{gathered} f\left(30\right)=-\frac{25}{3600}{\left(30\right)}^2+25 \\ f\left(30\right)=18.75ft \end{gathered}$$