Problem 31 - Finite Mathematics and Applied Calculus

Question

I Nutrition Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no vitamin C. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of vitamin \(\mathrm{C}\) for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the RDA of vitamin \(\mathrm{C}\) for infants. \(^{32}\) The cereal costs \(10 \mathrm{~d} / \mathrm{serving}\), the dessert costs \(53 \mathrm{~d} /\) serving, and the juice costs \(27 \mathrm{c} /\) serving. If you want to provide your child with at least 120 calories and at least 120 percent of the RDA of vitamin \(\mathrm{C}\), how can you do so at the least cost?

Step-by-Step Solution

Answer

By serving your child \(1\) serving of Gerber Apple Banana Juice, you will meet at least 120 calories and at least 120 percent of the RDA for vitamin C, at the least cost of 27 cents.

Step 1: Define the Variables

Let x be the number of servings of Gerber Mixed Cereal for Baby (\(60\) calories, \(0\%\) vitamin C, \(10\) cents/serving). Let y be the number of servings of Gerber Mango Tropical Fruit Dessert (\(80\) calories, \(45\%\) vitamin C, \(53\) cents/serving). Let z be the number of servings of Gerber Apple Banana Juice (\(60\) calories, \(120\%\) vitamin C, \(27\) cents/serving).

Step 2: Write the Inequalities for Calories and Vitamin C

The minimum requirements are \(120\) calories and \(120\%\) vitamin C. We can then write the following system of inequalities: 1. Calorie requirement: \(60x + 80y + 60z \geq 120\) 2. Vitamin C requirement: \(0x + 45y + 120z \geq 120\)

Step 3: Find the Cost Equation

The total cost of the servings can be represented by the equation: Cost \(= 10x + 53y + 27z\) (in cents)

Step 4: Solve the System of Inequalities and Minimize Cost

First, simplify inequality 1 by dividing by \(60\): \(x + \frac{4}{3}y + z \geq 2\) Now, consider inequality 2: \(45y + 120z \geq 120\) Since we are looking for the lowest cost and serving non-negative integers of food, we can analyze the possible scenarios: Scenario 1: \(1\) serving of Gerber Apple Banana Juice - Vitamin C requirement is met (120%) - Calorie requirement: \(60 \text{ calories}\) Scenario 2: \(1\) serving of Gerber Mango Tropical Fruit Dessert and \(2\) servings of Gerber Mixed Cereal for Baby - Vitamin C requirement is met (45%) - Calorie requirement: \(80 + 2(60) = 200 \text{ calories}\) Comparing both scenarios, we can see that scenario 1 meets both the calorie and vitamin C requirements and costs less. The Gerber Apple Banana Juice costs \(27\) cents/serving.

Step 5: Conclusion

By serving your child \(1\) serving of Gerber Apple Banana Juice, you will meet at least 120 calories and at least 120 percent of the RDA for vitamin C, at the least cost of 27 cents.

Key Concepts

Systems of Inequalities

In finite mathematics, a common application for systems of inequalities is to model constraints in real-life scenarios. A system of inequalities consists of multiple inequalities that simultaneously impose conditions on the variables involved.

Let's apply this concept to an example relevant to our dietary scenario—imagine you're planning meals for a child. Each food has a certain nutritional profile and cost, and you want to provide the child with a well-rounded diet without overspending. Just like in our problem concerning the cereal, dessert, and juice, we must satisfy at least two nutritional constraints (calories and vitamin C) while also considering the cost.

Mathematically, the inequalities would look something like this:

• 60x + 80y + 60z ≥ 120 for calories
• 0x + 45y + 120z ≥ 120 for vitamin C

Each inequality represents a 'boundary' that the meal plan cannot cross. We need to find combinations of servings (x, y, z) that satisfy both conditions at once. This is an essential step in seeking an optimal diet plan.

Optimal Solution

An optimal solution in the context of a system of inequalities is the best possible solution within the defined constraints. It must satisfy all of the inequalities in the system and, in the case of a minimization problem, result in the lowest possible value of the objective function—in our case, cost.

In the given nutrition problem, an optimal solution would provide the required 120 calories and 120 percent daily vitamin C, at the minimum cost. Optimal solutions are critical in real-world problems such as production, logistics, diet planning, and finance, as they ensure the most efficient use of resources. The process of identifying the optimal solution typically involves examining various feasible solutions and selecting the one that best fits the objective–cost-effectiveness in our dietary example.

Cost Minimization

Cost minimization is a fundamental goal in many optimization problems. It entails finding the least expensive way to achieve a certain objective while meeting all necessary requirements. As the cost is of primary concern, this typically involves formulating a cost function, which in our dietary example is represented by the equation:

Cost = 10x + 53y + 27z

Our goal is to minimize this cost function while still providing the nutritional needs of the child. In businesses, cost minimization might involve reducing production costs or finding more efficient ways to deliver services without compromising quality. In each case, careful analysis helps in determining how to achieve objectives at the lowest expense.

Nutritional Requirements

Nutritional requirements, such as daily calorie intake or vitamin C levels, are specific needs that must be met to maintain health. These requirements often serve as constraints in dietary optimization problems.

In the context of our example, the task was to meet or exceed the daily recommended allowance of vitamin C and a minimum calorie count. We must be mindful that these nutritional standards are set based on health and dietary research, so meeting these minimums is crucial for well-being. When determining meal plans, dietitians, or in our case, parents, must balance dietary requirements against other constraints like cost, making nutritional requirement constraints an integral part of many optimization problems in health and well-being.