Problem 25 - Finite Mathematics and Applied Calculus

Question

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of carbohydrates. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 21 grams of carbohydrates. \({ }^{11}\) If the cereal costs \(30 \phi\) per serving and the dessert costs 50 per serving, and you want to provide your child with at least 140 calories and at least 32 grams of carbohydrates, how can you do so at the least cost? (Fractions of servings are permitted.)

Step-by-Step Solution

Answer

To provide a child with at least 140 calories and 32 grams of carbohydrates at the least cost, you should buy about \(1.25\) servings of Gerber Mixed Cereal for Baby and about \(0.25\) servings of Gerber Mango Tropical Fruit Dessert. This will cost around \(48.75\phi\) and meet the calorie and carbohydrate requirements.

Step 1: Assign variables to the number of servings of each product

Let \(x\) be the number of servings of Gerber Mixed Cereal for Baby. Let \(y\) be the number of servings of Gerber Mango Tropical Fruit Dessert.

Step 2: Create inequalities to represent the calorie and carbohydrate requirements

Since each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of carbohydrates, and each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 21 grams of carbohydrates, we can write inequalities representing the calorie and carbohydrate requirements: Calorie requirement: \(60x + 80y \geq 140\) Carbohydrate requirement: \(11x + 21y \geq 32\)

Step 3: Create the cost function

The cost function represents the total cost of Gerber Mixed Cereal for Baby and Gerber Mango Tropical Fruit Dessert servings, with the cereal costing \(30 \phi\) per serving and the dessert costing 50 per serving: Cost function: \(C(x, y) = 30\phi x + 50y\)

Step 4: Graph the inequalities and find the feasible region

To graph the inequalities, we'll find the boundary lines for each requirement by replacing the inequality signs with equal signs: Boundary line for the calorie requirement: \(60x + 80y = 140\) Boundary line for the carbohydrate requirement: \(11x + 21y = 32\) Graph the two boundary lines and shade the feasible region (where both inequalities hold true, and \(x\) and \(y\) are non-negative).

Step 5: Find the minimum cost in the feasible region

Next, we'll use the cost function to minimize the cost in the feasible region (the intersection of the two shaded constraint inequalities and the positive x- and y-axis). Check the boundary lines and corner points of the feasible region to find the minimum cost. Calculate the cost function \(C(x, y)\) at those points. The point with the lowest cost function value will be the solution.

Step 6: Interpret the results

The solution will give the number of servings of Gerber Mixed Cereal for Baby (x) and Gerber Mango Tropical Fruit Dessert (y) that satisfy both the calorie and carbohydrate requirements with the least cost. Note that the minimum cost could occur at a fractional number of servings.

Key Concepts

Optimization Problems

Optimization problems are all about finding the best solution from a set of possible choices. In this case, the problem involves deciding on the number of servings of two food products, Gerber Mixed Cereal and Mango Tropical Fruit Dessert, to meet specific nutritional needs at the lowest cost. This is a classic example of a linear optimization problem where the solution involves minimizing a cost function while satisfying a set of requirements, called constraints.

The main goal here is to find the combination of servings that provides at least 140 calories and 32 grams of carbohydrates at the lowest possible cost. Linear programming and optimization problems often involve using graphical methods and calculations to find the point where the cost is minimized while still meeting all the specified requirements. The intersection of these constraints in a graph shows the feasible region, which is the set of all possible solutions that satisfy all conditions. The optimal solution will lie at one of the corners of this feasible region. This is because, in linear programming, the optimal solution of a linear function over a feasible region often lies at the vertices.

Inequalities

Inequalities are expressions stating that one value is larger or smaller than another, and they play a vital role in forming the constraints of an optimization problem. In this exercise, we deal with two main inequalities: one for calories and another for carbohydrates.

- **Calorie Inequality:** It ensures that the total calories from the chosen servings are at least 140. The inequality is written as: \[ 60x + 80y \geq 140 \] Here, 60 and 80 are the calories per serving for cereal and dessert, respectively.- **Carbohydrate Inequality:** This inequality ensures that the total carbohydrates from these servings are at least 32. It is expressed as: \[ 11x + 21y \geq 32 \] Where 11 and 21 are the grams of carbohydrates per serving for cereal and dessert, respectively. When dealing with inequalities, the solution process involves graphing these inequalities and identifying the overlapping region, known as the feasible region, where both conditions are met. This region represents all possible combinations of servings that meet the nutritional requirements.

Cost Functions

A cost function is a mathematical equation used to determine the total cost given the quantity of products consumed. In this example, you want to minimize the total cost while meeting minimum nutritional requirements. The cost function for this problem is:\[ C(x, y) = 30\phi x + 50y \]
This equation indicates that each serving of Gerber Mixed Cereal costs 30\(\phi\), and each serving of Mango Tropical Fruit Dessert costs 50. Here, \( x \) and \( y \) represent the number of servings for cereal and dessert, respectively.

By substituting different values of \( x \) and \( y \) that fall within the feasible region identified by the inequalities, you can calculate the total cost for each combination. The aim is to find the values of \( x \) and \( y \) that make this cost function as small as possible. This typically involves evaluating the cost function at the vertices of the feasible region where the inequality constraints meet or cross. By assessing these points, you can find the optimal solution that meets all requirements at the least cost.