Part $2$: Travel Demand Forecasting Problem

Due to the construction of several mega apartment complexes just south of campus, a developer is considering the construction of a new movie theater complex, "MegaMovies", on the east side of Gainesville. The developer has hired you to help determine the viability of building this theater complex. The cost of roadway improvements to the developer is a function of the amount of traffic being generated by the theater as well as the routes that these vehicles use in getting to$/$from the theater. Thus, you need to determine traffic demand on the available routes between the housing area and the proposed theater site.

There are three potential routes from this housing area to the proposed theater site. These routes have the following speed and length characteristics:

$\backslash$table$[[$Route$,$Free$-$Flow Speed $($mi$/$h$),$Length $($mi$)],[1,45,5\mathrm{.}5],[2,40,3\mathrm{.}5],[3,35,3\mathrm{.}75]]$

It is known that the individual route travel times increase $($in units of minutes$)$ according to the following functions $($with $x$ in units of $1000$ vehicles per hour$)$:

$\backslash$table$[[$Route$,\backslash$table$[[$Travel Time increase,$],[$as a function of traffic volume$]]],[1,0\mathrm{.}5x_1$