Assume you are conducting a demand modeling study for the morning commuting trips during the peak hour of a typical work day between the City of Bellevue and the City of Seattle. Below is some basic demographics information of the two cities $($these are factious numbers$)$:

$\backslash$table$[[,$City of Bellevue,City of Seattle$],[$# of households,$7,397,19,963],[$Total # of workers,$10,637,20,304],[$mean household income $($$$),111,728,49,110],[$# of establishments,$136,128]]$

The distance between the two cities is about $13\mathrm{.}4$ miles.

$(3)$ If a Gravity Model is to be used to estimate the trip distribution between Bellevue and Seattle, estimate the number of trips between the two cities based on the following assumptions $(d_{ij}$ is the distance between the two cities, in miles$)$:

$$ i is $0\mathrm{.}1$

$j$ is $0\mathrm{.}2$

$F_{ij}(d_{ij})=\frac{100}{d_{ij}^2}$

$K_{ij}=0\mathrm{.}01$

$(4)$ Available transportation modes between the two cities include: driving, transit, or bicycling. The average driving time between the two cities is $25$ minutes, with cost $($gasoline prices$)$ $$4\mathrm{.}00.$ Average transit time is $50$ minutes with a fare of $$0\mathrm{.}75.$ The distance of bicycling is: $15$ miles.

The following utilities functions are found to be appropriate to capture the choice behavior of travelers when determining which mode to choose.

Utility of driving: $U_d=2\mathrm{.}1-0\mathrm{.}25(co{s}_d)-0\mathrm{.}03(T{T}_d)$

Utility of transit: $Ut=1\mathrm{.}5-0\mathrm{.}25(co{s}_t)-0\mathrm{.}03(T{T}_t)$

Utility of bicycling: $Ub=1\mathrm{.}0-0\mathrm{.}1(D_b)$

Here costs in the above formula are in dollars, and travel times $($TT$)$ are in minutes. $D_b$ denotes the distance of bicycling, in miles.

Estimate the number of travelers using each travel mode by assuming travelers follow the discrete$-$choice principle.