91Ó°ÊÓ

The Gateway Arch in St. Louis is often mistaken to be parabolic in shape. In fact, it is a catenary, which has a more complicated formula than a parabola. The Arch is 625 feet high and 598 feet wide at its base.

Let $V=\left(0,625\right)$, since the arch is hanging $625ft$down from the horizontal and any point

$Q=\left(299,0\right)$is 299 ft away from the center since the arch is 598 ft wide.

The equation of the parabola opening upward is role="math" localid="1648217239962" ${\left(x-h\right)}^2=4a\left(y-k\right)$

substitute $h=0,k=625,x=299,y=0$in the equation of parabola

$$\begin{gathered} {\left(x-h\right)}^2=4a\left(y-k\right) \\ {\left(299-0\right)}^2=4a\left(0-625\right) \\ a=\frac{{299}^2}{4×625} \\ a=35.760 \end{gathered}$$

Substitute $a=35.760$in the equation of parabola

role="math" localid="1648218687236" $$\begin{gathered} x^2=4\left(35.760\right)\left(y-625\right) \\ {x}^2=143.04y-89400 \\ {x}^2+89400=143.04y \\ y=\frac{x^2+89400}{143.04} \end{gathered}$$

Substitute $x=\frac{567}{2}=283.5$in the equation of the parabola $y=\frac{x^2+89400}{143.04}$

$$\begin{gathered} y=\frac{x^2+89400}{143.04} \\ y=1186.88 \end{gathered}$$

Substitute $x=\frac{478}{2}=239$in the equation of the parabola role="math" localid="1648219519296" $y=\frac{x^2+89400}{143.04}$

$$\begin{gathered} y=\frac{x^2+89400}{143.04} \\ y=1024.33 \end{gathered}$$

Substitute $x=\frac{308}{2}=154$in the equation of the parabola $y=\frac{x^2+89400}{143.04}$

$$\begin{gathered} y=\frac{x^2+89400}{143.04} \\ y=790.79 \end{gathered}$$

Comparing the height of the given table and the height calculated as per the equation we conclude that the data does not support the notion that the Arch is in the shape of a parabola