Question

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 20 units of calcium, 10 units of iron, and 15 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 20 units of vitamin \(\mathrm{B}\). The minimum daily requirements for the diet are 400 units of calcium, 250 units of iron, and 220 units of vitamin 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 18 ounces of food \(\mathrm{X}\) and \(3.5\) ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{X}\) 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

A system of inequalities that the dietitian can use in the diet is \(20X + 15Y \geq 400\), \(10X + 20Y \geq 250\) and \(15X + 20Y \geq 220\). Any combination of X and Y within the common area enclosed by these inequalities will meet the minimum daily requirements.

Step-by-step solution

Construct the inequalities

Analyze the given information, we know that each ounce of food X contains 20 units of calcium, food Y contains 15 units and we need at least 400 units. This results in the first inequality: \(20X + 15Y \geq 400\).Repeat this process for iron and vitamin B, obtaining: \(10X + 20Y \geq 250\) and \(15X + 20Y \geq 220\).

Graph the system of inequalities

For convenience, simplify each inequality to slope-intercept form \(Y = MX + B\), then graph the lines: \[ Y \geq \frac{400 - 20X}{15}, Y \geq \frac{250 - 10X}{20}, Y \geq \frac{220 - 15X}{20} \] The common area the lines enclose represents the possible combinations of X and Y.

Find alternative combinations of foods X and Y

With supplies of food X running low, we need to find an alternative combination that still meets the daily requirements. Start from the original combination, 18 ounces of food X and 3.5 ounces of food Y, and traverse along the common area we identified previously. Any point in this region would constitute a valid combination that meets the daily requirements

Key concepts

Linear Programming

Linear programming is a mathematical method used for finding the best outcome in a given model. Its applications span from business and economics to engineering and the social sciences, particularly where systems are constrained by linear relationships. The essence of this approach relies on formulating a linear equation to maximize or minimize some quantities: profit, cost, or, as in our example, nutritional content.

In the context of nutritional mathematics, linear programming helps dietitians and nutritionists design meal plans that fulfill nutritional requirements while considering resource constraints. Here's how it's done:

• Define the objective function, which is the sum we aim to optimize.
• Construct linear inequalities based on resource limits and nutritional content per unit of food.
• Identify the feasible region, the set of all possible combinations of foods that meet the dietary constraints.
• Finally, find the optimal point(s) within the feasible region that maximizes or minimizes the objective function.

The diet supplement problem exemplifies linear programming where the aim is to meet or exceed daily nutritional requirements with limited food supplies.

Inequality Graphing

Graphing inequalities involves plotting regions on a graph that represent all possible solutions to an inequality. Unlike equations which are expressed as lines on a graph, inequalities are represented by shaded areas that include all the points that satisfy the inequality.

To graph a system of inequalities, follow these steps:

• First convert each inequality into slope-intercept form (if necessary).
• Graph the line for each inequality. Use a solid line if the inequality includes the equal sign (\(\geq\) or \(\leq\)), and a dashed line if it does not (\(>\) or \(<\)).
• Decide which side of the line to shade by choosing a test point (not on the line) and seeing if it satisfies the inequality.
• The intersection of the shaded areas - the area that is shaded for all inequalities in the system - is the solution set.

In the diet supplement problem, the graph provided a visual tool to see the feasible combinations of foods X and Y that satisfy all nutritional needs. Graphing inequalities is an integrative part of the linear programming process and is useful for visually understanding complex problems.

Nutritional Mathematics

Nutritional mathematics is a niche but practical application of mathematical concepts within the field of diet and nutrition. This specialized area leverages mathematical tools to plan diets, balance nutrient intake, and ensure food safety. In this context, linear programming and inequality graphing play pivotal roles in translating the dietary problem into a solvable mathematical model.

The objective is to achieve a balanced diet - one that meets all nutritional requirements without excess or deficiency of nutrients. Dietitians can use systems of inequalities to represent the conditions that the amounts of each food must meet:

• The constraints reflect the minimum daily requirements of nutrients, such as calcium, iron, and vitamin B.
• Each food item contributes certain amounts of each nutrient, creating a linear relationship.
• The goal is to find combinations of food that meet or exceed these nutrient requirements, falling within the feasible region identified by the inequalities.

By formulating a dietary problem as a system of linear inequalities, nutritional mathematicians can provide multiple options, ensuring that a balanced intake is accessible even with changes in food supply, as demonstrated in the diet supplement exercise.