This problem can be solved using linear programming, where we aim to maximize the total amount of money Donna can raise under time constraints. Let's break it down step by step.

Step 1: Define Variables

Let:

• $x$ represent the number of church groups Donna contacts.
• $y$ represent the number of labor unions Donna contacts.

Step 2: Define the Objective Function

The objective is to maximize the amount of money Donna raises, which is $150x + 175y$. This is the total money raised, where each church group raises $150 and each labor union raises $175.

Thus, the objective function is: $\text{Maximize } Z = 150x + 175y$

Step 3: Define the Constraints

Donna has constraints on the available time for letter writing and follow-up:

• Each church group requires 2 hours of letter writing, and each labor union requires 2 hours of letter writing. Donna can spend a maximum of 16 hours per month on letter writing. So, the constraint is: $2x + 2y \leq 16$ Simplifying this: $x + y \leq 8 \quad \text{(Letter writing constraint)}$

• Each church group requires 1 hour of follow-up, and each labor union requires 3 hours of follow-up. Donna can spend a maximum of 12 hours per month on follow-up. So, the constraint is: $x + 3y \leq 12 \quad \text{(Follow-up constraint)}$

Additionally, the number of groups Donna contacts cannot be negative: $x \geq 0, \quad y \geq 0$

Step 4: Graph the Feasible Region

We need to graph the system of inequalities to find the feasible region:

1. $x + y \leq 8$
2. $x + 3y \leq 12$
3. $x \geq 0$
4. $y \geq 0$

The feasible region is the area that satisfies all these inequalities.

Step 5: Find the Corner Points

The corner points of the feasible region are where the constraints intersect. Solving the system of equations, we get:

1. Set $x + y = 8$ and $x + 3y = 12$.

   Subtract the first equation from the second: $(x + 3y) - (x + y) = 12 - 8$ $2y = 4 \quad \Rightarrow \quad y = 2$

   Substitute $y = 2$ into $x + y = 8$: $x + 2 = 8 \quad \Rightarrow \quad x = 6$

   So, one corner point is $(6, 2)$.

2. Now, check the other points where the lines intersect with the axes:

   • If $x = 0$, from $x + 3y = 12$, we get $y = 4$. So, the point is $(0, 4)$.
   • If $y = 0$, from $x + y = 8$, we get $x = 8$. So, the point is $(8, 0)$.

The corner points are $(6, 2)$, $(0, 4)$, and $(8, 0)$.

Step 6: Evaluate the Objective Function at the Corner Points

Now we substitute these corner points into the objective function $Z = 150x + 175y$ to find the maximum.

• At $(6, 2)$: $Z = 150(6) + 175(2) = 900 + 350 = 1250$

• At $(0, 4)$: $Z = 150(0) + 175(4) = 0 + 700 = 700$

• At $(8, 0)$: $Z = 150(8) + 175(0) = 1200 + 0 = 1200$

Step 7: Conclusion

The maximum value of $Z$ occurs at $(6, 2)$, where Donna contacts 6 church groups and 2 labor unions, raising a total of $1250.

Final Answer:

• Donna should contact 6 church groups and 2 labor unions to raise the most money.
• The maximum amount of money she can raise is $1250.

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Do you have any questions or would you like more details? Here are 5 related questions to expand this topic:

1. How would the solution change if Donna had more follow-up hours available?
2. What happens if the amount raised from each church group or labor union changes?
3. Can the same method be applied if Donna had more than two types of groups to contact?
4. How would the problem change if some