Question

worthington hills grocery store has five regular checkout lines and one express line (12 items or less). based on a sampling study, it takes 15 minutes on the average for a customer to go through the regular line and 4 minutes to go through the express line. the store is open from 9 a.m. to 9 p.m. daily.
a. what is the stores maximum capacity (customers processed per day)? round your answer down to the nearest whole number.
customers processed per day
b. what is the stores capacity by day of the week if the five regular checkout lines operate according to the schedule below (the express line is always open)? do not round intermediate calculations. round your answers down to the nearest whole number.
hours/day mon tue wed thur fri sat sun
9 - 12 a.m. 1 2 1 2 2 4 3
12 - 4 p.m. 2 1 3 2 4 5 4
4 - 6 p.m. 4* 4 3 3 5 4 2
6 - 9 p.m. 4 3 4 3 5 4 1
*a \3\ means three regular checkout lines are open on monday from 4 to 6 p.m.
the stores capacity by day of the week (customers per day):
hours mon tue wed thur fri sat sun
totals

Explanation:

Step1: Calculate daily operating hours

The store is open from 9 a.m. to 9 p.m., so it is open for 12 hours a day.

Step2: Calculate capacity of regular lines per day

There are 5 regular lines and it takes 15 minutes per customer on average. In 1 hour (60 minutes), each regular - line can process $\frac{60}{15}=4$ customers. In 12 hours, 5 regular lines can process $5\times4\times12 = 240$ customers.

Step3: Calculate capacity of express line per day

There is 1 express line and it takes 4 minutes per customer on average. In 1 hour (60 minutes), the express - line can process $\frac{60}{4}=15$ customers. In 12 hours, the express line can process $1\times15\times12=180$ customers.

Step4: Calculate total maximum capacity per day

The total maximum capacity is the sum of the capacity of regular lines and express line, so $240 + 180=420$ customers.

Step5: Calculate capacity by day of the week

For each day of the week:
- **Monday**:
- Calculate capacity of regular lines:
- For 9 - 12 a.m. (3 hours), 1 line is open, processes $1\times\frac{60}{15}\times3 = 12$ customers.
- For 12 - 4 p.m. (4 hours), 2 lines are open, processes $2\times\frac{60}{15}\times4 = 32$ customers.
- For 4 - 6 p.m. (2 hours), 3 lines are open, processes $3\times\frac{60}{15}\times2 = 24$ customers.
- For 6 - 9 p.m. (3 hours), 4 lines are open, processes $4\times\frac{60}{15}\times3 = 48$ customers.
- Total for regular lines on Monday is $12 + 32+24 + 48=116$ customers.
- Capacity of express line on Monday is $\frac{60}{4}\times12 = 180$ customers.
- Total capacity on Monday is $116+180 = 296$ customers.
- **Tuesday**:
- For 9 - 12 a.m. (3 hours), 2 lines are open, processes $2\times\frac{60}{15}\times3 = 24$ customers.
- For 12 - 4 p.m. (4 hours), 1 line is open, processes $1\times\frac{60}{15}\times4 = 16$ customers.
- For 4 - 6 p.m. (2 hours), 4 lines are open, processes $4\times\frac{60}{15}\times2 = 32$ customers.
- For 6 - 9 p.m. (3 hours), 3 lines are open, processes $3\times\frac{60}{15}\times3 = 36$ customers.
- Total for regular lines on Tuesday is $24 + 16+32 + 36=108$ customers.
- Capacity of express line on Tuesday is 180 customers.
- Total capacity on Tuesday is $108 + 180=288$ customers.
- **Wednesday**:
- For 9 - 12 a.m. (3 hours), 1 line is open, processes $1\times\frac{60}{15}\times3 = 12$ customers.
- For 12 - 4 p.m. (4 hours), 3 lines are open, processes $3\times\frac{60}{15}\times4 = 48$ customers.
- For 4 - 6 p.m. (2 hours), 3 lines are open, processes $3\times\frac{60}{15}\times2 = 24$ customers.
- For 6 - 9 p.m. (3 hours), 4 lines are open, processes $4\times\frac{60}{15}\times3 = 48$ customers.
- Total for regular lines on Wednesday is $12 + 48+24 + 48=132$ customers.
- Capacity of express line on Wednesday is 180 customers.
- Total capacity on Wednesday is $132+180 = 312$ customers.
- **Thursday**:
- For 9 - 12 a.m. (3 hours), 2 lines are open, processes $2\times\frac{60}{15}\times3 = 24$ customers.
- For 12 - 4 p.m. (4 hours), 2 lines are open, processes $2\times\frac{60}{15}\times4 = 32$ customers.
- For 4 - 6 p.m. (2 hours), 3 lines are open, processes $3\times\frac{60}{15}\times2 = 24$ customers.
- For 6 - 9 p.m. (3 hours), 3 lines are open, processes $3\times\frac{60}{15}\times3 = 36$ customers.
- Total for regular lines on Thursday is $24+32 + 24+36 = 116$ customers.
- Capacity of express line on Thursday is 180 customers.
- Total capacity on Thursday is $116+180 = 296$ customers.
- **Friday**:
- For 9 - 12 a.m. (3 hours), 2 lines are open, processes $2\times\frac{60}{15}\times3 = 24$ customers.
- For 12 - 4 p.m. (4 hours), 4 lines are open, processes $4\times\frac{60}{15}\times4 = 64$ customers.
- For 4 - 6 p.m. (2 hours), 5 lines are open, processes $5\times\frac{60}{15}\times2 = 40$ customers.
- For 6 - 9 p.m. (3 hours), 5 lines are open, processes $5\times\frac{60}{15}\times3 = 60$ customers.
- Total for regular lines on Friday is $24+64 + 40+60 = 188$ customers.
- Capacity of express line on Friday is 180 customers.
- Total capacity on Friday is $188+180 = 368$ customers.
- **Saturday**:
- For 9 - 12 a.m. (3 hours), 4 lines are open, processes $4\times\frac{60}{15}\times3 = 48$ customers.
- For 12 - 4 p.m. (4 hours), 5 lines are open, processes $5\times\frac{60}{15}\times4 = 80$ customers.
- For 4 - 6 p.m. (2 hours), 4 lines are open, processes $4\times\frac{60}{15}\times2 = 32$ customers.
- For 6 - 9 p.m. (3 hours), 4 lines are open, processes $4\times\frac{60}{15}\times3 = 48$ customers.
- Total for regular lines on Saturday is $48+80 + 32+48 = 208$ customers.
- Capacity of express line on Saturday is 180 customers.
- Total capacity on Saturday is $208+180 = 388$ customers.
- **Sunday**:
- For 9 - 12 a.m. (3 hours), 3 lines are open, processes $3\times\frac{60}{15}\times3 = 36$ customers.
- For 12 - 4 p.m. (4 hours), 4 lines are open, processes $4\times\frac{60}{15}\times4 = 64$ customers.
- For 4 - 6 p.m. (2 hours), 2 lines are open, processes $2\times\frac{60}{15}\times2 = 16$ customers.
- For 6 - 9 p.m. (3 hours), 1 line is open, processes $1\times\frac{60}{15}\times3 = 12$ customers.
- Total for regular lines on Sunday is $36+64 + 16+12 = 128$ customers.
- Capacity of express line on Sunday is 180 customers.
- Total capacity on Sunday is $128+180 = 308$ customers.

Answer:

a. 420
b.
Mon: 296
Tue: 288
Wed: 312
Thur: 296
Fri: 368
Sat: 388
Sun: 308