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
See:
http://www.jiskha.com/display.cgi?id=1255637242
To determine the most profitable mixture of groups Jayanta should contact and the most money she can raise in a month, we can set up inequalities based on the given information.
Let's assume:
x = the number of church groups to contact
y = the number of labor unions to contact
Based on the time constraints, we can set up the following inequalities:
For letter writing time:
2x + 2y ≤ 16 (each church group requires 2 hours, each labor union requires 2 hours)
For follow-up time:
1x + 3y ≤ 12 (each church group requires 1 hour, each labor union requires 3 hours)
Since Jayanta wants to maximize the money raised, we can set up the objective function:
Profit = 100x + 200y (each church group raises $100, each labor union raises $200)
Now, we have the following system of inequalities:
2x + 2y ≤ 16
1x + 3y ≤ 12
We also have the objective function:
Profit = 100x + 200y
To find the most profitable mixture and the maximum money she can raise, we need to graph these inequalities on a coordinate plane and find the feasible region (where the inequalities are satisfied), then evaluate the profit function at each corner point within the feasible region to find the maximum profit.
Alternatively, you can solve this system of linear inequalities using linear programming methods such as the simplex method or graphical method in software like Excel solver, MATLAB, or online linear programming calculators. These methods will automatically find the optimal solution for you.
I hope this explanation helps you set up the inequalities and find the most profitable mixture for Jayanta!