Question 1165321
This is a system of linear inequalities problem, but since the problem asks to "find how many ounces of each type of food should be used... so that the minimum requirements... are met," without an objective function (like minimizing cost), we are looking for the **feasible region** that satisfies all constraints for both patients simultaneously.

Let:
* $A$ = ounces of **Food A** used in the meal.
* $B$ = ounces of **Food B** used in the meal.
* $C$ = ounces of **Food C** used in the meal.

We need to set up the nutritional constraints for Susan and Tom.

### 1. Set Up Constraints

The contents table summarizes the nutrients per ounce:

| Food | Calcium (mg/oz) | Iron (mg/oz) | Vitamin C (mg/oz) |
| :---: | :---: | :---: | :---: |
| A | 30 | 1 | 2 |
| B | 25 | 1 | 5 |
| C | 20 | 2 | 4 |

#### A. Susan's Constraints

| Nutrient | Requirement | Inequality |
| :---: | :---: | :---: |
| **Calcium** | $\ge 330$ mg | $30A + 25B + 20C \ge 330$ |
| **Iron** | $\ge 18$ mg | $1A + 1B + 2C \ge 18$ |
| **Vitamin C** | $\ge 44$ mg | $2A + 5B + 4C \ge 44$ |

#### B. Tom's Constraints

| Nutrient | Requirement | Inequality |
| :---: | :---: | :---: |
| **Calcium** | $\ge 280$ mg | $30A + 25B + 20C \ge 280$ |
| **Iron** | $\ge 13$ mg | $1A + 1B + 2C \ge 13$ |
| **Vitamin C** | $\ge 34$ mg | $2A + 5B + 4C \ge 34$ |

#### C. Non-Negativity Constraints
$$A \ge 0, \quad B \ge 0, \quad C \ge 0$$

### 2. Identify the Overall Feasible Region

Since the meal must meet **both** patients' requirements, the overall system is the *union* of the more restrictive constraints from Susan and Tom.

For example, if Susan needs $\ge 330$ mg of Calcium and Tom needs $\ge 280$ mg, the final meal must satisfy the more restrictive condition: $\ge 330$ mg.

| Nutrient | Susan's Minimum | Tom's Minimum | **Governing Constraint** |
| :---: | :---: | :---: | :---: |
| **Calcium** | $330$ | $280$ | $30A + 25B + 20C \ge 330$ |
| **Iron** | $18$ | $13$ | $A + B + 2C \ge 18$ |
| **Vitamin C** | $44$ | $34$ | $2A + 5B + 4C \ge 44$ |

The meal must satisfy the following system of linear inequalities:

1.  $$30A + 25B + 20C \ge 330$$
2.  $$A + B + 2C \ge 18$$
3.  $$2A + 5B + 4C \ge 44$$
4.  $$A, B, C \ge 0$$

### 3. Finding a Specific Solution (Example)

Since no objective function (like cost) was given to minimize, the solution is any set of non-negative values for $(A, B, C)$ that satisfies all three governing inequalities. The feasible region is a three-dimensional region (a polyhedron).

To provide a concrete answer, we can look for corner points (where the inequalities become equalities), as these often represent efficient solutions in optimization problems. However, without minimization, we'll look for a simple integer solution that works.

Let's test $A=5, B=5, C=5$:
1. $30(5) + 25(5) + 20(5) = 150 + 125 + 100 = 375 \ge 330$. (OK)
2. $5 + 5 + 2(5) = 5 + 5 + 10 = 20 \ge 18$. (OK)
3. $2(5) + 5(5) + 4(5) = 10 + 25 + 20 = 55 \ge 44$. (OK)

The meal mixture of **5 ounces of Food A, 5 ounces of Food B, and 5 ounces of Food C** meets both patients' minimum requirements.

**Final Answer:**

The feasible region is defined by the following system of inequalities:
$$\begin{cases} 30A + 25B + 20C \ge 330 \\ A + B + 2C \ge 18 \\ 2A + 5B + 4C \ge 44 \\ A \ge 0, B \ge 0, C \ge 0 \end{cases}$$

One possible solution is to use:
$$\mathbf{A = 5 \text{ ounces}, B = 5 \text{ ounces}, C = 5 \text{ ounces}}$$