Question

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 12 units of calcium, 10 units of iron, and 20 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 12 units of vitamin B. The minimum daily requirements for the diet are 300 units of calcium, 280 units of iron, and 300 units of vitamin \(\mathrm{B}\). (a) Find a system of inequalities describing the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that the dietitian can use in the diet. (b) Sketch the graph of the system. (c) A nutritionist normally gives a patient 10 ounces of food \(\mathrm{X}\) and 12 ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{Y}\) are running low. What other combinations of foods \(\mathrm{X}\) and \(\mathrm{Y}\) can be given to the patient to meet the minimum daily requirements?

Short answer

The system of inequalities that describes the different amounts of food X and Y is \(12x + 15y \geq 300\) (for calcium), \(10x + 20y \geq 280\) (for iron), and \(20x + 12y \geq 300\) (for vitamin B). After sketching the graph, a variety of combinations of food X and Y can be used to meet the diet requirements, particularly those with higher amounts of food X in response to the limited supply of food Y.

Step-by-step solution

Define Variables and Formulate Inequalities

Let's define the variables: x for the ounces of food X and y for the ounces of food Y. From the given units in each type of food we can then create inequalities to represent the daily requirements: \nFor calcium: \(12x + 15y \geq 300\) \nFor iron: \(10x + 20y \geq 280\) \nFor vitamin B: \(20x + 12y \geq 300\)

Sketch the Graph

To sketch the graph, one will need to express y in terms of x for each of the inequalities from Step 1 and plot on a graph. The region of overlap, where all inequalities hold true, represents the set of possible configurations of food X and Y that meet the necessary dietary requirements.

Finding Alternative Combinations

The intersection area represents all the combinations of food X and Y that meet the requirements. Since supplies of food Y are running low, one will need to seek alternative combinations in that region with less reliance on food Y, so combinations where X is higher but still within the feasibility region. These will need to be calculated from the boundaries set by the graphs in the previous step.

Key concepts

Linear Programming

Linear programming is a mathematical method used for finding the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. It is an optimization technique that can help in decision-making by maximizing or minimizing a linear objective function, subject to a set of linear inequalities called constraints.

In the context of the provided dietary supplement problem, the objective could be to minimize the cost of the diet while meeting nutritional requirements. The variables in this case would be the ounces of foods X and Y. The constraints would be derived from the minimum daily requirements of calcium, iron, and vitamin B, which lead to developing a system of linear inequalities based on these requirements. Linear programming methods, such as the Simplex Algorithm or Graphical Method, would then be used to find the optimal solution within the feasible region.

Graphing Inequalities

Graphing inequalities involves plotting on a coordinate system to show all the solutions of an inequality. When dealing with systems of linear inequalities, like those in the exercise, each inequality divides the graph into two regions: where the inequality is satisfied, and where it isn't. The area where it is satisfied is typically shaded.

The graphing process starts by transforming each inequality into an equation to draw the corresponding line. For the dietary problem, we convert the calcium, iron, and vitamin B inequalities into equations and plot them on a graph. The region where all the shaded areas overlap is the solution to the system of inequalities. It represents the combinations of foods X and Y that meet all daily nutritional requirements. Understanding how to graph these lines and determine the feasible region is crucial for solving linear programming problems.

Dietary Optimization

Dietary optimization is an application of linear programming where the goal is to create the most effective diet plan that meets nutritional requirements at the lowest cost or with other specified constraints. In real-life scenarios, dietary optimization takes into account various factors such as nutritional content, dietary restrictions, and budget limitations.

For the dietary exercise, we use the system of inequalities to identify the mix of foods X and Y to ensure that the patient receives the necessary nutrients. The constraints are the minimum units of calcium, iron, and vitamin B needed. By graphing these inequalities and identifying the feasible region, the dietitian can determine the most optimal proportions of each food that fulfills the patient's dietary requirements, especially when considering limitations such as a shortage of food Y.