Problem $3(30$ points$)-$ User Equilibrium $-$ Effect of Decreased Traffic

Two routes connect an origin and a destination, and the flow is $16,000ve\frac{h}{h}.$ Route $1$ has a performance function $t_1=4+3x_1,$ and route $2$ has a function of $t_2=b+5x_2,$ with the x $\text{'}$s expressed in thousands of vehicles per hour and the t$\text{'}$s in minutes. The user$-$equilibrium flow on route $1$ is $9980ve\frac{h}{h}$ and it is known that both routes are used.

Q$1$: First, determine the free$-$flow travel time on route $2($the parameter $b$ in route $2\text{'}$s performance function$)$ and equilibrium travel times.

Q$2$: Second, if the population declines reduce the number of travelers at the origin, and the total origin$-$destination flow is reduced to $6000ve\frac{h}{h},$ determine user$-$equilibrium travel times and flows.

Problem $4(15$ points$)-$ System Optimization and the Impact of Autonomous Vehicles on Total Travel Time

Consider the initial conditions for the two roads in Problem $3.$ Suppose all vehicles become autonomous and are centrally routed. With autonomous vehicles, the performance functions become $t_1=3+0\mathrm{.}5x_1$ for route $1$ and $t_2=1\mathrm{.}6+x_2,$ for route $2$ with the x $\text{'}$s expressed in thousands of vehicles per hours and the t$\text{'}$s in minutes.

Q: If the autonomous vehicles are assigned in a system optimal manner to minimize vehicles hours, how many vehicle hours will be saved relative to the user equilibrium solutions for conventional vehicles shown in Problem $3?$

SOLVE PROBLEM $4$ ONLY $!!$