Math Problem Statement
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.
- Formulate a model for this problem.
- Use desmos to solve this model.
- Should Dogs and Buns consider hiring a sixth full-time employee? (why / why not?)
- Write a paragraph to Dogs and Buns with your suggestions.
Solution
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 represent the number of hot dogs produced per week.
- Let represent the number of hot dog buns produced per week.
Objective Function:
We want to maximize the profit, which is given by: where is the total profit.
Constraints:
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Flour Constraint: Each bun requires 0.1 pounds of flour, and the maximum flour available is 200 pounds per week. Simplifying:
-
Pork Constraint: Each hot dog requires pound of pork, and the maximum pork available is 800 pounds per week. Simplifying:
-
Labor Constraint: Each hot dog requires 3 minutes of labor, and each bun requires 2 minutes of labor. The total labor available is minutes per week.
-
Non-negativity Constraints:
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 at the corner points of the feasible region to find the maximum profit.
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Plot the constraints:
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Find the intersection points (corner points) of the feasible region.
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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:
- What are the specific corner points of the feasible region after plotting the constraints?
- What is the exact profit at each of these corner points?
- How much does it cost to hire an additional full-time employee?
- How would changes in pork or flour costs affect the optimal solution?
- 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.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Constraints
Formulas
Objective function: P = 0.88x + 0.33y
Flour constraint: 0.1y ≤ 200
Pork constraint: (1/4)x ≤ 800
Labor constraint: 3x + 2y ≤ 12000
Theorems
Feasible Region in Linear Programming
Corner Point Theorem
Suitable Grade Level
Grades 10-12
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