Question $1[15$ marks$]$

A toll bridge carries $4500$ veh$/$day$.$ The current toll is $$\frac{1\mathrm{.}25}{?}$ vehicle. Studies have shown that for each increase in toll of $25$ cents, the traffic volume will decrease by $400$ veh$/$day$.$ It is desired to increase the toll to a point where revenue will be maximized.

a$)$ Write the expression for travel demand on the bridge related to toll increase and current volume.

b$)$ Determine toll charge to maximize revenues.

c$)$ Determine traffic in veh$/$day after toll increase.

d$)$ Determine total revenue increase with new toll. $[8$ marks$]$

An individual is planning to take a $1280-km$ trip between two large cities. Three possibilities exist: air, rail, or auto. The person is willing to pay $$25$ for every hour saved in making the trip. The trip by air costs $$600$ and travel time is $6$ hours, by rail the cost is $$450$ and travel time is $16$ hours, and by auto the cost is $$200$ and travel time is $24$ hours.

a$)$ Which mode is the best choice?

b$)$ What factors other than cost might influence the decision regarding which mode to use? $[7$ marks$]$