A decorative gateway is to be built in the form of two parabolic arches, each with its vertex at the origin and drawn to the same scale.
Arch A opens upward and is 3 m wide at a height of 2 m above the base.
Arch B opens to the right and is 3 m wide at a horizontal distance of 2 m from the vertex.
(a) For Arch A, find the equation of the parabola and the length of its latus rectum.
(b) For Arch B, find the equation of the parabola and the length of its latus rectum.
(c) Based on the latus rectum lengths found above, determine which arch is inherently wider and quantify the difference in their widths.