Math  /  Algebra

QuestionMary's restaurant has 5 full-time cooks each getting $140\$ 140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50\$ 50 a night. You use the character FF for fulltime cooks and PP for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
What is the constraints equation for number of employee from 8:00 to 9:00 PM?
F+P1+P2+P38F+P 1+P 2+P 3 \geq 8 F+P1+P28F+P 1+P 2 \leq 8 F+D1+D2+D2>RF+D 1+D 2+D 2>R

Studdy Solution

STEP 1

What is this asking? How many full-time and part-time cooks are needed between 8:00 PM and 9:00 PM to meet the minimum staffing requirement, without exceeding the limit on part-timers, and given that full-timers work from 5:00 PM to 11:00 PM and part-timers work for 3 hours? Watch out! Don't mix up the different time intervals or forget that full-time cooks work the entire evening!
Also, remember the limit on the total number of part-timers.

STEP 2

1. Analyze Full-time Cooks
2. Analyze Part-time Cooks
3. Formulate the Constraint

STEP 3

We know there are **5** full-time cooks, and they all work from 5:00 PM to 11:00 PM.
This means they're *always* present, including the 8:00 PM to 9:00 PM slot.
So, we have a constant contribution of **5** full-time cooks during that hour.

STEP 4

Part-timers work for **3** hours.
To be present from 8:00 PM to 9:00 PM, they must have started at 6:00 PM, 7:00 PM, or 8:00 PM.
Let's call the number of part-timers starting at these times P1P_1, P2P_2, and P3P_3, respectively.

STEP 5

If P1P_1 part-timers start at 6:00 PM, they'll work from 6:00 PM to 9:00 PM, covering the 8:00 PM to 9:00 PM slot.

STEP 6

If P2P_2 part-timers start at 7:00 PM, they'll work from 7:00 PM to 10:00 PM, also covering the 8:00 PM to 9:00 PM slot.

STEP 7

If P3P_3 part-timers start at 8:00 PM, they'll work from 8:00 PM to 11:00 PM, *also* covering the 8:00 PM to 9:00 PM slot.

STEP 8

We need *at least* **8** cooks between 8:00 PM and 9:00 PM.
We already have **5** full-timers.
The part-timers starting at 6:00 PM (P1P_1), 7:00 PM (P2P_2), and 8:00 PM (P3P_3) will *all* be working during this time.

STEP 9

So, the total number of cooks during 8:00 PM to 9:00 PM is the sum of the full-timers and these part-timers: 5+P1+P2+P35 + P_1 + P_2 + P_3.

STEP 10

Since we need *at least* **8** cooks, our constraint becomes: 5+P1+P2+P385 + P_1 + P_2 + P_3 \geq 8

STEP 11

The constraint for the number of employees from 8:00 PM to 9:00 PM is 5+P1+P2+P385 + P_1 + P_2 + P_3 \geq 8.

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