Dogs and Buns is a food processing plant which manufactures both hot dogs and hot dog buns. They
grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog
bun requires 0.1 pounds of flour. They currently have a contract with Pork Plus, for 800 pounds of pork
product delivered each week. Each hot dog requires ¼ pound of pork product. All the other ingredients
in the hot dogs and hot dog buns are in plentiful supply. Finally, the labor force at Dogs and Buns
consists of 5 employees working full time (40 hours per week each). Each hot dog requires 3 minutes of
labor, and each hot dog bun requires 2 minutes of labor. Each hot dog yields a profit of $0.88 and each
bun yields a profit of $0.33.
Dogs and Buns would like to know how many hot dogs and how many hot dog buns they should produce
each week so as to achieve the highest possible profit.
1. Formulate a model for this problem.
2. Use desmos to solve this model.
3. Should Dogs and Buns consider hiring a sixth full-time employee? (why / why not?)
4. Write a paragraph to Dogs and Buns with your suggestions.


### 1. Formulating the Model

This problem can be formulated as a Linear Programming (LP) problem. We need to maximize the profit function subject to the constraints imposed by the resources available (flour, pork, and labor).

#### Variables:
- Let $ x $ represent the number of hot dogs produced per week.
- Let $ y $ represent the number of hot dog buns produced per week.

#### Objective Function:
We want to maximize the profit, which is given by:
\[
\text{Maximize } P = 0.88x + 0.33y
\]
where $ P $ is the total profit.

#### Constraints:
1. **Flour Constraint:** 
   Each bun requires 0.1 pounds of flour, and the maximum flour available is 200 pounds per week.
   \[
   0.1y \leq 200
   \]
   Simplifying:
   \[
   y \leq 2000
   \]

2. **Pork Constraint:** 
   Each hot dog requires $ \frac{1}{4} $ pound of pork, and the maximum pork available is 800 pounds per week.
   \[
   \frac{1}{4}x \leq 800
   \]
   Simplifying:
   \[
   x \leq 3200
   \]

3. **Labor Constraint:** 
   Each hot dog requires 3 minutes of labor, and each bun requires 2 minutes of labor. The total labor available is $ 5 \times 40 \times 60 = 12000 $ minutes per week.
   \[
   3x + 2y \leq 12000
   \]

4. **Non-negativity Constraints:**
   \[
   x \geq 0, \quad y \geq 0
   \]

### 2. Solving the Model Using Desmos

To solve this LP problem, you would plot the constraints on a graph and identify the feasible region. Then, you would evaluate the objective function $ P = 0.88x + 0.33y $ at the corner points of the feasible region to find the maximum profit.

1. **Plot the constraints:**
   - $ y \leq 2000 $
   - $ x \leq 3200 $
   - $ 3x + 2y \leq 12000 $

2. **Find the intersection points (corner points) of the feasible region.**
3. **Evaluate the objective function at each corner point.**

**Note:** You can use Desmos or any other graphing tool to visualize the feasible region and find the corner points.

### 3. Should Dogs and Buns Consider Hiring a Sixth Full-Time Employee?

After solving the problem using Desmos, you'll determine whether the labor constraint is binding (i.e., whether it is limiting the production and profit). If the labor constraint is limiting, then hiring an additional employee might allow Dogs and Buns to produce more hot dogs and buns, increasing their profit. However, this decision also depends on the cost of hiring the new employee and whether the additional profit outweighs this cost.

### 4. Suggestions for Dogs and Buns

**Dear Dogs and Buns Management,**

Based on the analysis of your production constraints and profit margins, the optimal production levels of hot dogs and buns have been determined. By producing the recommended quantities, you will achieve the highest possible profit under the current resource constraints. If labor is the limiting factor in your production, hiring an additional employee could potentially increase your production capacity and overall profit. However, this should be weighed against the cost of hiring. If the cost of an additional employee is lower than the profit gained from increased production, hiring would be beneficial.

I recommend revisiting your production strategy regularly, especially if there are changes in resource availability or labor costs.

Best regards,  
[Your Name]

### Follow-up Questions:
1. What are the specific corner points of the feasible region after plotting the constraints?
2. What is the exact profit at each of these corner points?
3. How much does it cost to hire an additional full-time employee?
4. How would changes in pork or flour costs affect the optimal solution?
5. Could automating part of the process reduce labor costs or constraints?

**Tip:** Regularly reassess your constraints to ensure they align with market conditions and resource availability.

Dogs and Buns is a food processing plant which manufactures both hot dogs and hot dog buns. They grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog bun requires 0.1 pounds of flour. They currently have a contract with Pork Plus, for 800 pounds of pork product delivered each week. Each hot dog requires ¼ pound of pork product. All the other ingredients in the hot dogs and hot dog buns are in plentiful supply. Finally, the labor force at Dogs and Buns consists of 5 employees working full time (40 hours per week each). Each hot dog requires 3 minutes of labor, and each hot dog bun requires 2 minutes of labor. Each hot dog yields a profit of $0.88 and each bun yields a profit of $0.33. Dogs and Buns would like to know how many hot dogs and how many hot dog buns they should produce each week so as to achieve the highest possible profit. 1. Formulate a model for this problem. 2. Use desmos to solve this model. 3. Should Dogs and Buns consider hiring a sixth full-time employee? (why / why not?) 4. Write a paragraph to Dogs and Buns with your suggestions.

Learn how to formulate and solve a linear programming problem to maximize profit in the production of hot dogs and buns. This problem involves optimizing a profit function subject to constraints on flour, pork, and labor resources. Suitable for high school students in Grades 10-12.

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