The required number of lanes.

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Chapter 6, Problem 37P

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The required number of lanes.

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Consider, M people (aka pax) who want to travel by car from O to D. They all start working at D at Q (e.g., Q=8am). If a person departs at time t, assume the time needed to go from O to D is given by c(t)=A+Bx(t), where x(t) is the flow of people departing at time t [car/unit of time]. In addition, a is the penalty for being early at work (E(t) is how early the person arrived when departing at time t), and ẞ is the penalty for being late at work (L(t) is how late the person arrived when departing at time t). Assume 0 < a < 1 < ß. Further assume the departure time choice problem under the equilibrium conditions. Prove that the arrival time of people who depart when most of the M people start their trips is equal to Q.

1. Plot the SWRC (suction vs. volumetric water content) from the van Genuchten (1980) model for the following “base-case” parameters (assume m = 1-1/nvG): alpha vG = 0.35 kPa-1, nvG = 2.2,Delta res = 0.02, and delta S = delta sat = 0.45. These values are approximately representative of a sand.Perform a sensitivity analysis on each of the base case parameters (i.e., vary each by ±10% while holding the other three constant) to determine their relative effects on the SWRC. For example, how would a change in s alter the curve if the other base case parameters stay the same? In your answer, provide 4 plots for the parametric evaluation of each parameter as well as a brief qualitative explanation for each plot (for example, when investigating the effect of alpha vG, show the base-case curve and the curves with different values of alpha vG on the same plot). Be sure that suction is plotted on a logarithmic scale. Also, discuss how theparameters of the van Genuchten SWRC model might differ…

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