t 0 5 10
r(t)0 90 0
A shop is open 9 AM - 7 PM. The function r(t), gives the rate at which customers arrive (in people/hour) at time t, where t measures time in hours since 9 AM. Suppose that the salespeople can serve customers at a rate of 75 people per hour. At 4 hrs after opening people begin having to wait in line before getting served, and the line is longest 6 hours after opening. Answer the following questions:
How many people are in the line at the time when it's the longest?
Answer: people.
When does the line vanish?
Answer: hours after opening.
To find the number of people in the line when it's the longest, we need to determine the surplus of customers arriving compared to the salespeople's capacity to serve them.
First, let's calculate the number of customers arriving in the time interval of 4 to 10 hours after opening:
t: 4 5 6 7 8 9 10
r(t): 0 90 180 165 90 0 0
Here, I have included the range of 4 to 10 hours as this includes the time when people begin having to wait in line until the time when the line is longest.
To find the number of customers in the line, we subtract the salespeople's capacity to serve from the rate of customers arriving:
Customers in line at t=4: 90 - 75 = 15
Customers in line at t=5: 180 - 75 = 105
Customers in line at t=6: 165 - 75 = 90
Customers in line at t=7: 90 - 75 = 15
Customers in line at t=8: 0 - 75 = -75 (not considered as negative)
Customers in line at t=9: 0 - 75 = -75 (not considered as negative)
Customers in line at t=10: 0 - 75 = -75 (not considered as negative)
From the calculations above, we see that the line is the longest when there are 105 people in it at t=5.
Next, to find when the line vanishes, we need to determine the time when the rate of customers arriving becomes equal to or lower than the salespeople's capacity to serve.
Looking at the values of r(t) from t=4 to t=10, we see that the rate of customers arriving becomes lower than 75 at t=9. After that, the arrival rate remains at zero.
Therefore, the line vanishes 9 hours after opening.
So, the answer to your questions are:
1. There are 105 people in the line when it's the longest.
2. The line vanishes 9 hours after opening.