Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A dietitian wishes to plan a meal around three foods. The meal is to include 8800 units of vitamin A, 3380 units of vitamin \(\mathrm{C}\), and 1020 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the following table: $$\begin{array}{lccc}\hline & \text { Food I } & \text { Food II } & \text { Food III } \\\\\hline \text { Vitamin A } & 400 & 1200 & 800 \\\\\hline \text { Vitamin C } & 110 & 570 & 340 \\\\\hline \text { Calcium } & 90 & 30 & 60 \\\\\hline\end{array}$$ Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements.

Step-by-Step Solution

Answer

The dietitian cannot meet the vitamin and calcium requirements with the given foods, as the system of linear equations derived from the problem has no solution.

Step 1: Define variables

Let's denote the number of ounces of each food as follows: - \(x\) = ounces of Food I - \(y\) = ounces of Food II - \(z\) = ounces of Food III

Step 2: Set up linear equations

Based on the problem description and the table given, we have the following linear equations: 1. \(400x + 1200y + 800z = 8800\) (Vitamin A requirements) 2. \(110x + 570y + 340z = 3380\) (Vitamin C requirements) 3. \(90x + 30y + 60z = 1020\) (Calcium requirements) The linear system is: $$ \begin{cases} 400x + 1200y + 800z = 8800 \\ 110x + 570y + 340z = 3380 \\ 90x + 30y + 60z = 1020 \end{cases} $$

Step 3: Solve the linear system using matrix methods

First, represent the system as an augmented matrix: $$ \left[\begin{array}{ccc|c} 400 & 1200 & 800 & 8800 \\ 110 & 570 & 340 & 3380 \\ 90 & 30 & 60 & 1020 \end{array}\right] $$ Next, perform Gaussian elimination (row reduction) to find the values of \(x\), \(y\), and \(z\): Add \(-\frac{11}{4}R_1 + R_2 \to R_2\) and \(-\frac{9}{4}R_1 + R_3 \to R_3\): $$ \left[\begin{array}{ccc|c} 400 & 1200 & 800 & 8800 \\ 0 & -210 & -140 & -850 \\ 0 & -270 & -180 & -1170 \end{array}\right] $$ Now, multiply \(R_2\) by \(-\frac{1}{210}\) and \(R_3\) by \(-\frac{1}{270}\): $$ \left[\begin{array}{ccc|c} 400 & 1200 & 800 & 8800 \\ 0 & 1 & \frac{2}{3} & 4 \\ 0 & 1 & \frac{2}{3} & \frac{13}{3} \end{array}\right] $$ Subtract \(R_2\) from \(R_3\) to eliminate the second variable in \(R_3\): $$ \left[\begin{array}{ccc|c} 400 & 1200 & 800 & 8800 \\ 0 & 1 & \frac{2}{3} & 4 \\ 0 & 0 & 0 & 1 \end{array}\right] $$ The last row of the matrix implies that \(0 = 1\), which is a contradiction. Thus, there is no solution for this system of linear equations, and it is not possible to meet the vitamin and calcium requirements with the given foods.

Key Concepts

Nutrient Optimization

In the real world, nutrient optimization refers to creating a diet plan that meets specific nutrient needs using a set of available food options. For dietitians, achieving this often involves ensuring that meals include an adequate amount of essential nutrients, such as vitamins and minerals, while also considering taste, cost, and availability. In our example, the dietitian wants to plan a meal that provides exact amounts of vitamin A, vitamin C, and calcium using three different foods. This requires careful calculation to ensure that each nutrient requirement is precisely met.

• Vitamin A target: 8800 units
• Vitamin C target: 3380 units
• Calcium target: 1020 units

Each food contributes differently to these nutrient needs. The challenge is to determine the right combination of foods, making nutrient optimization an important skill for effective meal planning.

System of Linear Equations

A system of linear equations is a set of equations with multiple variables. In our context, each equation corresponds to the nutritional requirement for a specific nutrient. The variables represent the quantities of each food needed to meet these requirements. With the given nutrients: vitamin A, vitamin C, and calcium, we need to solve a system of equations to figure out how much of each food is required.
The system is represented by:

• Vitamin A: \(400x + 1200y + 800z = 8800\)
• Vitamin C: \(110x + 570y + 340z = 3380\)
• Calcium: \(90x + 30y + 60z = 1020\)

Here, \(x\), \(y\), and \(z\) are the quantities of Food I, Food II, and Food III, respectively. Solving such systems is crucial to finding a feasible solution for meeting dietary needs.

Matrix Methods

Matrix methods provide a structured approach to solving systems of linear equations. By representing the equations in matrix form, we can use various techniques to simplify and solve them. This often involves creating an augmented matrix that includes both the coefficients of the variables and the constant terms from the equations.
For our nutrient optimization problem, the system of linear equations translates into an augmented matrix:\[\begin{bmatrix}400 & 1200 & 800 & | & 8800 \110 & 570 & 340 & | & 3380 \90 & 30 & 60 & | & 1020\end{bmatrix}\]Using matrix operations, such as row reductions, allows us to systematically find solutions to the variables \(x\), \(y\), and \(z\). Matrix methods are powerful for solving complex systems efficiently, especially when dealing with multiple variables.

Gaussian Elimination

Gaussian Elimination is a popular algorithm used to solve systems of linear equations. It involves transforming the system's augmented matrix into a row-echelon form to easily extract solutions to the variables. This step-by-step method uses row operations—a series of adding, subtracting, or scaling rows—to simplify matrices.
In our exercise, we applied Gaussian elimination to:

• Simplify the augmented matrix
• Isolate each variable step by step

Unfortunately, the elimination process led us to a contradiction: a row indicating \(0 = 1\). This shows the system of equations is inconsistent, meaning no combination of the available foods can meet all nutrient requirements simultaneously. Despite this, knowing Gaussian elimination is essential as it highlights potential inconsistencies and offers an algebraic understanding of solving linear systems.