An architecture firm is designing a parabolic entrance for a new community center. The arch is 10 feet wide at its base and reaches a maximum height of 5 feet. On the project's coordinate blueprints, the lower left side of the arch is placed at the origin $(0, 0)$. According to the standard modeling procedure for this arch, what are the coordinates of the vertex?

A structural engineering firm is developing a blueprint for a parabolic support beam that spans 10 feet at its base and reaches a maximum height of 5 feet. Arrange the following steps in the correct order to derive the mathematical equation for this specific arch design.

A structural design team is modeling a parabolic support arch that is 10 feet wide at its base and 5 feet high. To create the mathematical model, the team places the lower left end of the arch at the origin $(0, 0)$. Match each component of the arch's coordinate model to its correct value or point.

An architectural designer is modeling a parabolic entryway that is 10 feet wide and 5 feet high. Using a coordinate system where the bottom-left corner of the arch is at $(0, 0)$ and the vertex is at $(5, 5)$, what is the numerical value of the leading coefficient $a$ in the equation $y = a(x - 5)^2 + 5$? (Express your answer as a simplified fraction).

Logistics Warehouse Doorway Model