Many problems in industry, science, psychology, and the medical fields involve situations with quantities that must either be maximized or minimized. For example, businesses are interested in maximizing profits while minimizing costs and time. Microbiologists are interested in minimizing contaminants in a city water system while maximizing output and efficiency. The Red Cross may be interested in maximizing the number of bottled water and medical kits to earthquake survivors. All of the situations involve analyzing several variables and linear inequalities at once to find an optimal solution. For this assessment, your job is to:
1.	Identify a similar (possibly hypothetical) situation.
2.	Write an objective function describing the quantity that must be maximized or minimized.
3.	Identify and describe the constraints within the given situation.
4.	Write a system of 2–3 inequalities that describes these constraints.
5.	Graph the system of inequalities from part 4, using only the first quadrant and its boundary.
6.	Evaluate the objective function for maximum/minimum (depending on your situation) output.
7.	Write a statement identifying the values of each constraint that give the maximum/minimum output for your situation and interpret this maximum/minimum output within your given situation. Use language appropriate for your audience and career setting. For example, if you were writing a report to your immediate boss, the language in this statement should be clear, concise, and make sense for the situation.

The graph above shows the feasible region where the system of inequalities is satisfied. This region represents all the possible combinations of **Product A** and **Product B** production that meet the constraints of raw materials, labor hours, and production capacity.

### 6. Evaluating the Objective Function

To find the maximum profit, we evaluate the objective function $ P = 50x + 70y $ at the vertices (corner points) of the feasible region. The vertices can be found where the lines intersect:

1. **Intersection of $ 2x + 4y = 1000 $ and $ x + 3y = 600 $**.
2. **Intersection of $ x = 300 $ and $ 2x + 4y = 1000 $**.
3. **Intersection of $ x = 300 $ and $ x + 3y = 600 $**.

I'll calculate the coordinates of these vertices and evaluate the profit at each.

The vertices of the feasible region where the constraints intersect are all at the point $(300, 100)$. The profit at this point is:

\[
P = 50(300) + 70(100) = 22{,}000 \text{ dollars}
\]

### 7. Final Statement and Interpretation

**Maximum Profit Statement:**
To maximize profit, the company should produce 300 units of **Product A** and 100 units of **Product B**. This production plan will yield a maximum profit of $22,000. This solution fully utilizes the available labor hours and raw materials, while staying within production capacity limits.

Would you like any further details or explanations? Here are some related questions:

1. How do we determine the feasibility of different constraint combinations?
2. What would happen if the production capacity of **Product A** was increased?
3. How does changing the profit per unit affect the objective function?
4. Can you explain how each constraint impacts the shape of the feasible region?
5. How would additional constraints, such as a maximum budget, affect this analysis?
6. What tools can be used to solve such optimization problems in real-world scenarios?
7. How do we interpret infeasible solutions in a real-world context?
8. What if the objective was to minimize costs instead of maximizing profit?

**Tip:** When solving optimization problems, always check the boundaries of your feasible region as these often hold the key to finding the maximum or minimum value of the objective function.

Explore a detailed example of linear programming applied to maximize profit in business scenarios. Learn how to formulate an objective function and constraints using systems of inequalities. Evaluate the feasible region graphically and calculate maximum profit using vertex analysis.

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