Problem 48 - Elementary Algebra Within Reach

Question

A dietitian is asked to design a special dietary supplement using two different foods. Each ounce of food \(X\) contains 20 units of calcium, 15 units of iron, and 10 units of vitamin B. Each ounce of food Y contains 10 units of calcium, 10 units of iron, and 20 units of vitamin B. The minimum daily requirements in the diet are 280 units of calcium, 160 units of iron, and 180 units of vitamin B. Write a system of linear inequalities that describes the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that can be used in the diet.

Step-by-Step Solution

Answer

The system of linear inequalities that describes the different amounts of food X and Y that can be used in the diet is: \n20x + 10y ≥ 280, \n15x + 10y ≥ 160, \n10x + 20y ≥ 180.

Step 1: Identify the variables

Let's denote the ounces of food X as \(x\) and the ounces of food Y as \(y\).

Step 2: Transpose the Nutrient Information for Each Food Into Equations

Considering that each ounce of food \(X\) contains 20 units of calcium, 15 units of iron, and 10 units of vitamin B, and each ounce of food \(Y\) contributes 10 units of calcium, 10 units of iron, and 20 units of vitamin B, we'll turn this into expressions:\n\nFor Calcium: 20x + 10y\nFor Iron: 15x + 10y\nFor Vitamin B: 10x + 20y

Step 3: Formulate the Inequalities

We know that the body needs at least 280 units of calcium, 160 units of iron, and 180 units of Vitamin B, so we'll set up the following inequalities:\n\nCalcium: 20x + 10y ≥ 280\nIron: 15x + 10y ≥ 160\nVitamin B: 10x + 20y ≥ 180

Key Concepts

Dietary Supplement Optimization

When a dietitian sets out to create a balanced dietary supplement, the goal is to blend different foods in such a way that the combination meets specific nutritional requirements. This is a classic example of an optimization problem, which can be elegantly addressed using systems of linear inequalities.

To conceptualize this, imagine a dietitian is balancing two foods, each with its own profile of nutrients—food X and food Y. The objective is to find the least amount of each food that provides enough calcium, iron, and vitamin B, to satisfy the minimum daily requirements without over-consuming any single nutrient. By setting up a system of linear inequalities, the dietitian can visually map out the feasible region where the amounts of foods X and Y meet all the nutritional guidelines simultaneously.

This method of using inequalities, rather than equations, allows for flexibility in the solution. There is usually not just one exact amount of each food needed, but rather a range of viable combinations. Thus, the solution provides a variety of options, from which the dietitian can select an optimal mix based on additional factors like cost, availability, or consumer preference.

Algebraic Formulation of Real-World Problems

Translating real-world scenarios into algebraic expressions is a powerful tool for quantitative analysis and problem-solving. In our example, the dietitian deals with a real-world problem: determining the right mix of foods to meet nutritional requirements.

This process involves identifying variables and translating verbal descriptions into algebraic terms. For instance, the dietitian assigns the variables 'x' and 'y' to represent the ounces of food X and Y, respectively. The nutrient content of each food is then expressed algebraically, forming a foundation upon which to build a system of inequalities. In this case, the expressions 20x + 10y for calcium, 15x + 10y for iron, and 10x + 20y for vitamin B are devised by considering the amount of each nutrient per ounce of food.

Creating these algebraic expressions enables the dietitian to use mathematical techniques to analyze and solve for the amounts of each food needed. It turns subjective nutritional balancing into an objective mathematical operation, making the solution approach clear, precise, and verifiable.

Nutrient Requirement Constraints

In the realm of dietary supplement optimization, constraints play a crucial role. These constraints often take the form of nutrient requirement guidelines that must be met or exceeded. The dietary needs of the body, like minimum daily requirements for calcium, iron, and vitamin B, are perfect illustrations of such constraints.

In algebra, these nutritional guidelines are translated into inequalities because the diet must include at least, but not limited to, the specified amount of nutrients. For example, the system of linear inequalities for our nutritional problem—20x + 10y ≥ 280 for calcium, 15x + 10y ≥ 160 for iron, and 10x + 20y ≥ 180 for vitamin B—indicates the lowest limit of each nutrient that the combination of foods X and Y must provide.

These constraints define the feasible region, which is the set of all possible (x, y) combinations that satisfy all the inequalities. Meeting these nutrient requirement constraints ensures that the dietary supplement will support health objectives, and investigating the feasible region helps determine the most cost-effective and nutritionally adequate blend of foods.