can someone please help me set up inequalities based on the info below?:
jayanta is working to raise money for the homeless by sending information letters and making follow-up calls to local labor organizations and church gorups. She discovered that each church group requires 2 hr of letter writing and 1 hr of followup, while for each labor union she needs 2 hr of letter writing and 3 hr of followup. Jayanta can raise $100 from each church group and $200 from each union local, and she has a maximum of 16 hr of letter writing time and a maximum of 12 hr of follow-up time avaliable per month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month.
C = Church (2 hours Writing, 1 hour follow-up)
U = Labour Union (2 hours writing, 3 hours follow-up)
Constraints:
C ≥ 0
U ≥ 0
2C+2U ≤ 16
C+3U ≤ 12
Z = 100C+200U (objective function)
To set up the inequalities, let's assign variables to represent the number of church groups and labor unions Jayanta should contact.
Let:
- x be the number of church groups
- y be the number of labor unions
Now, we can set up the inequalities based on the given information:
1. Letter Writing Time:
Jayanta spends 2 hours of letter writing for each church group and 2 hours for each labor union. The total letter writing time is limited to 16 hours per month, so we have the inequality:
2x + 2y ≤ 16
2. Follow-up Time:
Jayanta spends 1 hour of follow-up for each church group and 3 hours for each labor union. The total follow-up time is limited to 12 hours per month, so we have the inequality:
1x + 3y ≤ 12
3. Non-negativity:
We cannot have negative numbers of church groups or labor unions, so we have the inequalities:
x ≥ 0
y ≥ 0
Now that we have set up the inequalities, we can find the most profitable mixture of groups and the maximum money Jayanta can raise in a month. Since Jayanta receives $100 from each church group and $200 from each labor union, we need to maximize the revenue:
Revenue = 100x + 200y
We can solve this system of inequalities and find the values of x and y that maximize the revenue using graphical or algebraic methods, such as substitution or elimination.