An amusement park has commissioned the design of a steel roller coaster with a drop section that is modeled by a parabola. Part of the roller coaster’s track will go through an underground tunnel. In this section, the roller coaster will dip [math]12[/math] feet below ground level. The roller coaster will dip below ground level at a horizontal distance of [math]24[/math] feet from the peak just before the drop and reemerge to ground level at a horizontal distance of [math]36[/math] feet from the peak just before the drop. Find an equation of the parabola that describes the drop and the height of the roller coaster at the peak.
[list=1] [*]Derive the coordinates of the points on the parabola. [*]Establish the linear factors and the equation of the parabola. [*]Find the vertex of the parabola. [*]Find the value of [math]a[/math]. [*]Find the height of the peak of this section of the roller coaster. [/list]