The City of Georgetown can spend a maximum of $800.00 of food for a Thank You Pot Luck. It cost $8 for a pound of meat and $5 for a pound of tofu. Write an inequality to represent the pound of meat, x, and tofu, y, you can purchase.   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.

### 1. Similar Hypothetical Situation

Let's consider a similar situation: A school is planning a fundraising event and can spend a maximum of $600 on decorations. The school plans to buy balloons and banners. Balloons cost $2 per pack, and banners cost $10 each. 

### 2. Objective Function

The objective function describes the total cost spent on the items and is to be minimized. For the original Georgetown situation, the objective function could be represented as:

\[ C(x, y) = 8x + 5y \]

where:
- $x$ represents the pounds of meat,
- $y$ represents the pounds of tofu.

### 3. Constraints

In this situation, the main constraints are:
- The total amount of money spent on meat and tofu cannot exceed $800.
- The quantities of meat and tofu purchased cannot be negative.

### 4. System of Inequalities

The constraints can be described by the following system of inequalities:

1. The budget constraint:
   \[ 8x + 5y \leq 800 \]

2. Non-negativity constraints:
   \[ x \geq 0 \]
   \[ y \geq 0 \]

### 5. Graph the System of Inequalities

To graph the system of inequalities, consider only the first quadrant since $x$ and $y$ represent pounds, which cannot be negative.

- Plot the line $8x + 5y = 800$. This line represents the boundary of the budget constraint.
- Shade the region below and to the left of the line $8x + 5y = 800$ to represent all possible combinations of $x$ and $y$ that satisfy the budget constraint.
- Ensure the region is in the first quadrant (where $x \geq 0$ and $y \geq 0$).

### 6. Evaluate the Objective Function

To find the maximum amount of meat and tofu that can be purchased:

- Identify the intercepts of the line $8x + 5y = 800$:
  - When $x = 0$, $5y = 800$ ⟹ $y = 160$ (maximum tofu with no meat).
  - When $y = 0$, $8x = 800$ ⟹ $x = 100$ (maximum meat with no tofu).
- Evaluate the objective function at these intercepts and any intersection points within the feasible region to find which provides the maximum quantity of either meat or tofu under the constraints.

### 7. Statement of Maximum Output

To maximize the quantity of food while staying within budget, Georgetown should consider the following options:

- **100 pounds of meat and 0 pounds of tofu:** This uses the entire $800 budget and maximizes the quantity of meat.
- **0 pounds of meat and 160 pounds of tofu:** This uses the entire $800 budget and maximizes the quantity of tofu.
- Any other combination that lies on the boundary line $8x + 5y = 800$ also uses the full budget, but the choice depends on the preference for the proportion of meat to tofu.

**Conclusion:** Georgetown has the flexibility to purchase up to 100 pounds of meat or 160 pounds of tofu, or any combination thereof, as long as the total cost does not exceed $800. Depending on the dietary preferences of the attendees, the city can decide the optimal mix within these constraints.

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Do you want more details or have any questions about this solution?

### Related Questions

1. How would the solution change if the budget was increased to $1,000?
2. What happens if the cost of meat decreases to $6 per pound?
3. How would you represent the situation if tofu was a fixed cost instead of variable per pound?
4. How can this situation be analyzed using linear programming?
5. What is the significance of the intercepts in the graph of the inequality?
6. How can you adjust the graph to account for changes in prices or budget?
7. Can this situation have more than one optimal solution? Why or why not?
8. What real-life factors might affect the decision beyond just cost?

### Tip

When graphing inequalities, always double-check which region represents the feasible solutions by substituting test points into the inequalities.

Explore how to optimize a food budget for a potluck event by maximizing the quantity of meat and tofu purchased under cost constraints. Learn about linear programming, inequalities, and graphing techniques applied to real-world budgeting problems.

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