Question

Consider the car-caravan analogy from Section 4 in Chapter 1. In this problem, assume a propagation speed of 80 km/hr and that each toll booth takes 6 seconds to service a car.
a) (Please Show Work) Suppose the caravan of 10 cars begins immediately in front of the first toll booth, travels 45 km to a second toll booth, then another 55 km to a third toll booth, and finally stops immediately after the third tool booth. Thus, they travel a total of 100 km. What is the total end-to-end delay?
b) (Please Show Work) Where is the last car in the caravan after one hour? Your answer must include a distance/specific location, and not only a relative direction.

Answer 1

Given : Propagation speed = 80 km/hr
Tollbooth delay = 6 seconds/car
Total distance traveled by caravan= 100 km
Total cars = 10
No. of tollbooths=3

a) Total end to end delay = ?
Total end to end delay = Tollbooth delay + Traveling time

Traveling time = Total distance traveled by cars / Propagation speed
= 100 km / 80 km/hr
= 1.25 hrs-----1

Tollbooth delay = Tollbooth delay of 1 car * No. of cars * No. of tolls
= 6 seconds/car * 10 * 3
= 180 seconds or 3 minutes-------2

Total end to end delay = ------1 + ------2
= 1.25 hrs + 3 min
= 75 min + 3 min
= 78 min or 1.3 hrs

b) Last car in caravan after 1 hour = ?

Total time-traveled by the last car = Total time-traveled - Tollbooth delay ----6

Tollbooth delay = No. of Tollbooths crossed * Total waiting time for the last car----3

As the third tollbooth is after 110 km(45km + 55km), the car can't reach till 3rd tollbooth
Because at the max speed of 80 km/hr the car can travel distance of 80 km in 1 hour

Total 2 tollbooths would be crossed in 1-hour journey, first and second------4
Total waiting time for the last car = Tollbooth delay of 1 car * No. of cars
= 6 seconds/car * 10
= 60 seconds-----5

From ----3,----4,----5
Tollbooth delay = 2 * 60 seconds
= 120 seconds ----7

From ---6 and ---7

Total time-traveled by the last car = Total time-traveled - Tollbooth delay
= 1 hr - 120 seconds
Converting in minutes
= 60 minutes - 2 minutes
= 58 minutes

Thus the last car travels for 58 minutes

Distance covered by the last car in 58 minutes = Propogation speed *Total time traveled
= 80 km/hr * 58 minutes
Convert minutes into hour by dividing it by 60
=80 km/hr * (58/60)
= 80 km/hr * 0.967
= 77.36 km

So the last car in the caravan is at 77.36km from the starting point in 1 hour