Q3(a) · Store-and-forward caravan

Q3(a). Ten cars queue at a tollbooth; service is one car per second. Each road to the next booth is 400 km; speed is 10 km/s. A booth waits for the entire caravan before servicing its first car. Find: (1) one-car service time; (2) whole-caravan service time; (3) first car travel time after departure; (4) last car travel time after departure; (5) time from first departure to its next service; (6) whether first and second booths ever service cars simultaneously; (7) whether there is an interval with neither servicing any car. [10 marks]

Answer

Given

10 cars; service1car/s; road400km; speed10km/s; each booth waits for all10 cars.

To find

All seven timing/overlap questions.

Formula and why

LaTeX: T_{service}=N/r,\quad T_{travel}=d/v

Transmission corresponds to serializing the whole caravan; propagation is the road journey of each individual car.

Solution

Step 1. Choose t=0 as first service start

Car1 leaves at1s; car10 leaves at10s. Travel=400/10=40s. Thus first arrival at booth2 is41s and last arrival is50s. Booth2 starts service only at50s.

Step 2. Answer every printed subpart

Part	Answer	Reason
1	1 second	One car per second
2	10 seconds	10/1
3	40 seconds	Measured from first car departure
4	40 seconds	Last car travels the same road at the same speed
5	49 seconds	From first departure t1 to next service t50:50−1
6	No	First booth service0–10; second50–60
7	Yes	Neither is servicing during(10,50); in particular before first arrival, (10,41)

Common mistake: part5 is not50s because its clock starts when the first car leaves, one second after the first service began.