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 ​$125 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 12 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
  
enter your response herex 1plus
  
enter your response herex 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.

This problem is a **linear programming** problem where the goal is to maximize the money raised by Donna. We'll break it into two parts as instructed.

### Part 1: Defining the Objective Function

Let:
- $ x_1 $ represent the number of church groups contacted.
- $ x_2 $ represent the number of labor unions contacted.

From the problem:
- Donna raises $125 from each church group, so the contribution from church groups is $ 125x_1 $.
- Donna raises $200 from each labor union, so the contribution from labor unions is $ 200x_2 $.

Thus, the **objective function** (the total amount of money raised, $ z $) is:
\[
z = 125x_1 + 200x_2
\]

This is the function Donna needs to maximize.

### Part 2: Constraints

Donna's time constraints are:
- She has a maximum of 16 hours for letter writing.
  - Each church group requires 2 hours of letter writing.
  - Each labor union also requires 2 hours of letter writing.
  
  So, the total letter writing time constraint is:
  \[
  2x_1 + 2x_2 \leq 16
  \]
  
- She has a maximum of 12 hours for follow-up.
  - Each church group requires 1 hour of follow-up.
  - Each labor union requires 3 hours of follow-up.

  So, the total follow-up time constraint is:
  \[
  x_1 + 3x_2 \leq 12
  \]

Also, since $ x_1 $ and $ x_2 $ represent the number of groups and unions, they must be non-negative:
\[
x_1 \geq 0 \quad \text{and} \quad x_2 \geq 0
\]

Now, the task is to maximize the objective function $ z = 125x_1 + 200x_2 $, subject to the constraints:
\[
\begin{aligned}
2x_1 + 2x_2 &\leq 16 \\
x_1 + 3x_2 &\leq 12 \\
x_1, x_2 &\geq 0
\end{aligned}
\]

### Solution: Graphical or Simplex Method

We can solve this problem using either the **graphical method** or the **simplex method**. Let me calculate the values that maximize $ z $.

Let me do the calculations for you.Donna should contact:
- **6 church groups** and
- **2 labor unions**

to raise a **maximum of $1,150** in donations.

Would you like to go over more details on the solution process or the methods used? Here are five questions to expand your understanding:

1. How does the simplex method work for solving linear programming problems?
2. What are the implications of the constraints on the final solution?
3. Could there be alternative ways to represent the constraints for different scenarios?
4. How would changes in available time affect the number of groups and unions contacted?
5. How could we incorporate additional factors like variable donation amounts?

**Tip:** When solving linear programming problems, always ensure your objective function and constraints are clearly defined before attempting any solution method.

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 $125 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 12 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.

This problem explores how Donna De Paul can maximize her fundraising efforts for the homeless using linear programming. With constraints on time and donations per group, learn how to determine the most profitable combination of church groups and labor unions to raise up to $1,150 per month. Suitable for Grade 12 or college-level students studying optimization.

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