Question

A building has an entry with the shape of a parabolic arch 66 feet high and 46 feet wide at the base. Find an equation for the parabola if the vertex is placed at the origin of the coordinate system.

Answer 1

So. hmmm we know the arc entry has a vertical clearance of 66 feet, and is 46feet wide

notice the picture below.. if we move the arc to the origin, we end up with a parabola like so, from -23, 0 to 23, 0

now  $\bf \boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\ x=a(y-{{ k}})^2+{{ h}}\qquad\qquad vertex\ ({{ h}},{{ k}})\\\\ -----------------------------\\\\ y=a(x-0)^2+0\implies y=ax^2 \\\\\\ \textit{what's the value of "a"? well, we know } \begin{cases} x=23\\ y=-66 \end{cases} \\\\\\ -66=a(23)^2\implies \cfrac{-6}{529}=a\qquad thus\qquad \boxed{y=-\cfrac{6}{529}x^2}$