Problem $2[10$ points$]$

A left turning movement has a maximum arrival rate of $400$ vph $.$ The saturation flow of this movement is $1,400vph.$ For this approach, the yellow time is $4$ seconds, all red time is $2$ seconds, and total lost time is $3$ seconds. The cycle time is $120$ seconds. The turn bay that is provided for the left turning vehicles is $200$ ft long.

a$)$ What minimum displayed green time must be provided to ensure that the queue does not exceed the turn bay length? Assume a vehicle length of $20$ ft $.($Hint: Use a queuing diagram for traffic approaching and passing the signal from the left turn bay$).$

b$)$ What is the average delay of left turning vehicles $($i$.$e$.,$ all left turning vehicles, not only the delayed ones$)?$

c$)$ What is the longest delay any of the left turning vehicles will experience? Show all work