Donna De Paul is raising money for the homeless. She discovers that each church group requires 2 hours of letter writing and 1 hour of​ follow-up, while for each labor union she needs 2 hours of letter writing and 3 hours of​ follow-up. Donna can raise ​$150 from each church group and ​$200 from each union​ local, and she has a maximum of 16 hours of letter writing and a maximum of 14 hours of​ follow-up available per month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month.
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Part 1
 Let x 1 be the number of church​ groups, and let x 2 be the number of labor unions. What is the objective​ function?
zequals
  
150x 1plus
  
200x 2
​(Do not include the​ $ symbol in your​ answers.)
Part 2
She should contact 
  
enter your response here church​ group(s) and 
  
enter your response here labor​ union(s), to obtain a maximum of ​$
  
enter your response here in donations.

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:
\[
\begin{aligned}
2x_1 + 2x_2 & \leq 16, \\
x_1 + 3x_2 & \leq 14, \\
x_1, x_2 & \geq 0.
\end{aligned}
\]

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.

Donna De Paul is raising money for the homeless. She discovers that each church group requires 2 hours of letter writing and 1 hour of follow-up, while for each labor union she needs 2 hours of letter writing and 3 hours of follow-up. Donna can raise $150 from each church group and $200 from each union local, and she has a maximum of 16 hours of letter writing and a maximum of 14 hours of follow-up available per month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month. Let x1 be the number of church groups, and let x2 be the number of labor unions. What is the objective function?

Learn how to solve a real-world optimization problem using linear programming. Donna De Paul aims to raise money for the homeless by contacting church groups and labor unions, with constraints on her available time. This problem covers key linear programming concepts, objective function, and constraints. Suitable for Grades 10-12.

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