The Waterfall Buffet in the lower level of the National Art Gallery serves food cafeteria-style daily to visitors and employees. The buffet is self-service. From 7:00 A.M. to 9:00 A.M. customers arrive at the buffet at a rate of 10 per minute; from 9:00 A.M. to noon, at 4 per minute; from noon to 2:00 P.M., at 14 per minute; and from 2:00 P.M. to closing at 5:00 P.M., at 8 per minute. All the customers take about the same amount of time to serve themselves from the buffet. Once a customer goes through the buffet, it takes 0.4 minute to pay the cashier. The gallery does not want a customer to have to wait longer than 4 minutes to pay. How many cashiers should be working at each of the four times during the day?
To determine the number of cashiers required at each time during the day, we need to calculate the maximum number of customers that can be served by a single cashier within a 4-minute window. We can then divide the total number of customers by this value to find the number of cashiers needed.
Let's calculate the number of customers each cashier can serve within 4 minutes.
From 7:00 A.M. to 9:00 A.M., the arrival rate is 10 customers per minute, and the time taken by each customer to serve themselves is the same.
The total number of customers served by a cashier in this period can be calculated by multiplying the arrival rate (10 customers/minute) by the time taken to serve each customer (4 minutes):
10 customers/minute * 4 minutes = 40 customers
From 9:00 A.M. to noon, the arrival rate is 4 customers per minute, and the time taken by each customer to serve themselves is the same.
The total number of customers served by a cashier in this period can be calculated by multiplying the arrival rate (4 customers/minute) by the time taken to serve each customer (4 minutes):
4 customers/minute * 4 minutes = 16 customers
From noon to 2:00 P.M., the arrival rate is 14 customers per minute, and the time taken by each customer to serve themselves is the same.
The total number of customers served by a cashier in this period can be calculated by multiplying the arrival rate (14 customers/minute) by the time taken to serve each customer (4 minutes):
14 customers/minute * 4 minutes = 56 customers
From 2:00 P.M. to 5:00 P.M., the arrival rate is 8 customers per minute, and the time taken by each customer to serve themselves is the same.
The total number of customers served by a cashier in this period can be calculated by multiplying the arrival rate (8 customers/minute) by the time taken to serve each customer (4 minutes):
8 customers/minute * 4 minutes = 32 customers
Now, let's calculate the number of cashiers needed at each time during the day by dividing the total number of customers by the number of customers served by a single cashier.
From 7:00 A.M. to 9:00 A.M.:
Number of cashiers = Total number of customers / Number of customers served by a single cashier
Number of cashiers = 40 customers / 40 customers = 1 cashier
From 9:00 A.M. to noon:
Number of cashiers = 16 customers / 16 customers = 1 cashier
From noon to 2:00 P.M.:
Number of cashiers = 56 customers / 56 customers = 1 cashier
From 2:00 P.M. to 5:00 P.M.:
Number of cashiers = 32 customers / 32 customers = 1 cashier
Therefore, you would need 1 cashier at each of the four times during the day.