Q $3)$ Solve the example problem shown in Figure of an intersection. If the speed$-$dens ty relationship is linear and flow states A and $C$ are defined as follows: $UA$ and $uC$ are equal to $40$ and $26$ miles per hour, respectively, and $kA$ and $kc$ are equal to $24$ and $60$ vehicles per mile per lane, respectively. Assume a $30-$second green phase.

$[20]$

Plot a graph of the stopping shock wave $($WAB$)$ in Figure, as a function of the arrival flow $($qA$)-$Assume a linear speed$-$density relationship of $u=50-0\mathrm{.}417k.$

Plot a graph of the starting shock wave $($wBC$)$ in Figure as a function of the discharge flow $($qc$).$ Assume a linear speed$-$density relationship $u=50-0\mathrm{.}417k.$

Calculate the minimum length of the cycle that will not result in queue spillover to the next signal cycle.

$($a$)$