Problem $3[8$ points$]$

Two major cities, A and B $,$ are located $440$ miles apart $($like the distance from Boston to Washington, D$.$C$.),$ and currently the only economical way to travel between the two cities is to fly. A high$-$speed rail service is being proposed.

The scheduled $($one$-$way$)$ flight time by commercial airliner is $1\mathrm{.}25$ hours, but a trip also requires on average $1$ hour to get from $A$ to the origin airport, $1$ hour at the airport for security checks and waiting to board, and then $1$ hour to get from the destination airport to B$.$ The typical fare for a one$-$way ticket between A and B is $$50.$

A high speed rail line has been proposed to run non$-$stop trains between the cities at an average speed of $160$ mph $.$ The train stations will be located in the center of each city, so the average time to access the terminal is now estimated to be half that of accessing the airport $($i$.$e$.,0\mathrm{.}5$ hours from $A$ to the origin train station, and $0\mathrm{.}5$ hours from the destination station to the B$).$ Trains can be boarded immediately before departure, so there is no additional time associated with waiting at the station. The one$-$way ticket price for the train, $F,$ has yet to be determined.

A preliminary demand study determines that the average utility for a business traveler is a function of in$-$vehicle travel time $($IVT$),$ out$-$of$-$vehicle travel time $($OVT$),$ access time $(ACC),$ and ticket price $($COST$)$ for the round$-$trip:

$V=x+{}_{\text{I}\text{V}\text{T}}$IVT$+{}_{\text{O}\text{V}\text{T}}(OVT+$ACC$)+{}_{\text{C}\text{O}\text{S}\text{T}}$COST

In the above, $x$ denotes the economic benefit a traveler receives from a round$-$trip between A and $B$ and is the same for all travelers. From the demand study, the estimated coefficient values are ${}_{\text{I}\text{V}\text{T}}=-0\mathrm{.}3,{}_{\text{O}\text{V}\text{T}}=-0\mathrm{.}5,$ and ${}_{\text{C}\text{O}\text{S}\text{T}}=-0\mathrm{.}01.$

a$)$ Using the estimated coefficients for the preferences of the average business traveler, determine the utilities of the airline and rail options.

b$)$ According to the logit model, what percent of the business trips is the train expected to capture if the round$-$trip train tickets are priced the same as airline tickets? At what train ticket price $F$ will the mode share $($for the train$)$ be $50\%?$ Show all work