We can solve this problem using linear programming, where the goal is to maximize the amount Donna can raise given her time constraints.

Part 1: Objective Function

The amount of money Donna raises depends on the number of church groups $(x_1)$ and labor unions $(x_2)$ she contacts. From each church group, she raises $150, and from each labor union, she raises $200. So the objective function (which represents the total money raised) is:

$z = 150x_1 + 200x_2$

This is the function Donna wants to maximize.

Part 2: Constraints

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

1. Letter writing constraint: Each church group requires 2 hours, and each labor union requires 2 hours of letter writing. She has a maximum of 16 hours available for letter writing: $2x_1 + 2x_2 \leq 16$

2. Follow-up constraint: Each church group requires 1 hour, and each labor union requires 3 hours of follow-up. She has a maximum of 14 hours available for follow-up: $x_1 + 3x_2 \leq 14$

3. Non-negativity constraint: Donna cannot contact a negative number of church groups or labor unions: $x_1 \geq 0, \quad x_2 \geq 0$

Solution

We now need to solve this linear programming problem by finding the values of $x_1$ and $x_2$ that maximize $z = 150x_1 + 200x_2$, subject to the constraints.

Let’s summarize the system of inequalities:

2x_1 + 2x_2 & \leq 16, \\ x_1 + 3x_2 & \leq 14, \\ x_1, x_2 & \geq 0. \end{aligned}$$ We can solve this graphically or by using the simplex method to find the optimal values of $$x_1$$ and $$x_2$$. Let's calculate the maximum values for $$x_1$$ and $$x_2$$. #### Solving the system From the first constraint: $$2x_1 + 2x_2 \leq 16 \quad \text{or} \quad x_1 + x_2 \leq 8.$$ This gives a simplified constraint. From the second constraint: $$x_1 + 3x_2 \leq 14.$$ Now we can solve this system. The optimal solution to maximize the amount of money Donna can raise is as follows: - She should contact **5 church groups** and **3 labor unions**. - The maximum amount of money she can raise in a month is: $$z = 150(5) + 200(3) = 750 + 600 = 1350$$ Thus, Donna can raise **$1350** in total. ### Key Summary: - Objective function: $$z = 150x_1 + 200x_2$$ - Maximum number of church groups to contact: 5 - Maximum number of labor unions to contact: 3 - Maximum total amount of money raised: $1350 Would you like a more detailed explanation of how these values were determined? --- Here are 5 related questions for further understanding: 1. How would the solution change if Donna had more hours available for follow-up? 2. What if Donna could raise more money from labor unions, say $250 per union? 3. How would the objective function change if Donna received a bonus for contacting both types of groups? 4. What if the number of hours available for letter writing and follow-up were reduced? 5. Could a different combination of groups raise more money if the constraints were relaxed? **Tip:** Linear programming is especially useful for optimizing resource allocation, like maximizing profit or minimizing cost, subject to constraints.