***Only need parts iii), iv), & vii)***

Question B: Car traffic arrives at a large toll plaza at different rates depending on the time of day. The rate (in cars-per-hour) at time t is r(t) = 5000*(1 - 0.8cos(2 pi t / 24)) for t measured in hours between 0 and 24.

i) Define A(T) = the # of cars that have arrived by time T. Find a formula for A(T).

ii) The maximum departure rate is 500 cars/hour per lane, with 15 lanes, for a total maximum of 7500 cars/hour. At what time does the arrival rate start to exceed that, so that the tollbooths cant serve all the cars immediately and a queue starts building up?

iii) From that time point you identified in part (ii), the # of cars that have departed the tollbooth system by time T is called D(T), and it increases linearly at a rate of 7500 cars/hr until it intersects A(T) again--that is when the queue has finally dwindled down to 0. Find a formula for D(T) between those two timepoints (when the queue starts and when it dwindles down to 0)

iv) Find the area between the A(T) and D(T) curves in that time interval. This gives the total # of car*hours spent waiting in the queue, which is also the number of person*hours if each car has only 1 person. This is the total time wasted in the tollbooth backup for that day.

v) What is the meaning of A(13)-D(13) ?

vi) Imagine a horizontal line segment starting at (13,A(13)) and ending where it touches the D(T) curve, at roughly 13.644. What is the meaning of the length of that line segment?

vii) Divide your answer from part (iv) by the total # of cars that arrived in that time interval, to get the average delay per car.