Problem 70 - Precalculus with Limits

Question

A dietitian designs 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 of the diet are 300 units of calcium, 150 units of iron, and 200 units of vitamin B. (a) Write a system of inequalities describing the different amounts of food X and food Y that can be used. (b) Sketch a graph of the region corresponding to the system in part (a). (c) Find two solutions of the system and interpret their meanings in the context of the problem.

Step-by-Step Solution

Answer

The inequalities that describe the situation are: \(20X + 10Y \geq 300\), \(15X + 10Y \geq 150\), and \(10X + 20Y \geq 200\), with \(X \geq 0\) and \(Y \geq 0\). The graph would be a two-dimensional plot with the lines representing these inequalities. The feasible region will be where all three regions from the inequalities overlap and quantities are non-negative. The two solutions chosen in the feasible region represent dietary combinations that satisfy the nutritional needs.

Step 1: Define the Variables

Let \(X\) and \(Y\) represent the ounces of food X and Y respectively.

Step 2: Formulate the Inequalities

Based on the nutrient contents and requirements, the inequalities will be as follows:\n\nCalcium: \(20X + 10Y \geq 300\) \nIron: \(15X + 10Y \geq 150\) \nVitamin B: \(10X + 20Y \geq 200\) \nAll quantities cannot be negative, so \(X \geq 0\) and \(Y \geq 0\)

Step 3: Sketch the Graph

Sketch a graph where \(X\) and \(Y\) are on the x-axis and y-axis respectively. Plot the lines \(20X + 10Y = 300\), \(15X + 10Y = 150\), and \(10X + 20Y = 200\). The region of interest is where all three inequalities overlap, and \(X\) and \(Y\) are non-negative.

Step 4: Find Solutions

Select two points in the region of interest; these represent different combinations of foods X and Y that meet all nutritional requirements. For instance, one point could be the intersection of the calcium and vitamin B lines, and another could be the intersection of the iron and vitamin B lines (ensure both points are in the feasible region).

Step 5: Interpret the Results

Each point will describe a dietary mix in ounces of foods X and Y that meet the nutritional requirements. The exact combinations will depend on the points chosen in the previous step.

Key Concepts

Dietary Supplement Formulation

Creating an effective dietary supplement requires careful consideration of nutritional content and how different food combinations can meet dietary needs. In the exercise presented, a dietitian is designing a supplement with two foods, X and Y, which have different nutritional profiles.

Formulating a dietary supplement begins with defining the problem: the minimum daily requirements. Each food has a certain amount of calcium, iron, and vitamin B. To ensure the supplement meets the daily requirements for these nutrients, the dietitian needs to find the right balance between food X and food Y.

The challenge is identifying the ounces of foods X and Y that will collectively satisfy these nutritional needs without being wasteful or excessive. This is where the concept of systems of inequalities comes into play – it allows the dietitian to outline all possible combinations of X and Y that are adequate. The process is mathematically rigorous, ensuring objectivity in the diet supplement formulation.

Nutritional Requirements Problem

Nutritional requirements are the cornerstone of diet planning and supplement creation. In the context of our problem, the task involves ensuring that the consumption of foods X and Y leads to at least 300 units of calcium, 150 units of iron, and 200 units of vitamin B, which represent the defined minimum daily requirements.

This problem requires setting up systems of inequalities that represent these constraints:

• Calcium: must have at least 20 units from X and 10 units from Y per ounce.
• Iron: needs a minimum of 15 units from X and 10 units from Y per ounce.
• Vitamin B: requires no less than 10 units from X and 20 units from Y per ounce.

Defining the variables properly and setting up these inequalities correctly is vital. It ensures no nutritional deficit while considering that an ounce of either food can't be negative, which makes sense practically since one can't have a negative quantity of food.

Graphical Method in Linear Programming

The graphical method is an essential technique used in linear programming to solve problems involving two variables, like our nutritional requirements problem. It provides a visual representation of the feasible region defined by systems of inequalities.

In the sketched graph for our problem, each inequality is represented by a line on a graph with food X on the x-axis and food Y on the y-axis. Where these lines intersect defines the feasible region—combinations of X and Y that meet all nutritional requirements.

By plotting these lines and identifying the overlap area, one can visually inspect suitable mixes of foods X and Y. The corner points of the feasible region, where the lines intersect, often represent optimal solutions, provided they meet all the constraints. By choosing different points within this region, we can propose various combinations of our foods that fulfill the dietary supplement's requirements.