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 ​$175 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 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**.

---

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

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 $175 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.

Discover how to use linear programming to solve an optimization problem involving church groups and labor unions. This problem explores key concepts such as objective functions, constraints, and corner point evaluation. Suitable for students in Grades 10-12, this step-by-step solution helps maximize profits in fundraising scenarios.

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