Problem 27 - Beginning and Intermediate Algebra

Question

A hospital dietician must plan a lunch menu that provides 485 Cal, 41.5 g of carbohydrates, and 35 mg of calcium. A 3 -oz serving of broiled ground beef contains \(245 \mathrm{Cal}, 0 \mathrm{g}\) of carbohydrates, and 9 mg of calcium. One baked potato contains \(145 \mathrm{Cal}, 34 \mathrm{g}\) of carbohydrates, and 8 mg of calcium. A one-cup serving of strawberries contains 45 Cal, 10 g of carbohydrates, and 21 mg of calcium. (Source: Home and Garden Bulletin No. 72, U.S. Government Printing Office, Washington, D.C. 20402 ) How many servings of each are required to provide the desired nutritional values?

Step-by-Step Solution

Answer

To meet the overall nutritional requirements, the servings required are approximately 1.542 servings of broiled ground beef, 0.191 servings of baked potato, and 3.5 servings of strawberries. However, since fractional servings of food items are impractical, rounding up to the nearest whole number and adjusting the menu accordingly is advised.

Step 1: Variables

Let's define the variables: - Let 'x' represent servings of broiled ground beef - Let 'y' represent servings of baked potato - Let 'z' represent servings of strawberries Now, let's set up linear equations based on the provided nutritional information.

Step 2: Calories Equation

The first equation will represent the total calories requirement and thus, \(245x + 145y + 45z = 485\) (Calories)

Step 3: Carbohydrates Equation

The second equation will represent the total carbohydrates requirement, so: \(0x + 34y + 10z = 41.5\) (Carbohydrates)

Step 4: Calcium Equation

The third equation will represent the total calcium requirement, which is: \(9x + 8y + 21z = 35\) (Calcium) Now, we have a system of 3 linear equations: 1. \(245x + 145y + 45z = 485\) 2. \(0x + 34y + 10z = 41.5\) 3. \(9x + 8y + 21z = 35\)

Step 5: Solve the System of Equations

To solve this system of linear equations, you can use either substitution, elimination, or a matrix method. In this case, we will use the matrix method to find the solution. First, let's arrange the coefficients into an augmented matrix: ``` | 245 145 45 | 485 | | 0 34 10 | 41.5| | 9 8 21 | 35 | ``` Now, we will perform Gaussian elimination to reduce the matrix to a triangular form. After completing the Gaussian elimination, we will obtain the following resulting matrix: ``` | 245 145 45 | 485 | | 0 34 10 | 41.5 | | 0 0 70 | 245 | ``` From here, we can now use back substitution to solve for the variables x, y, and z: For the third equation, we have \(70z = 245\). By solving for z, we obtain \(z = 3.5\). Now, we substitute the value of z back into the second equation which is \(34y+10z=41.5\). By substituting the value of z, we can solve for y: \(34y + 10(3.5) = 41.5 \rightarrow 34y = 6.5\) ⟹ \(y = 0.191\) Finally, we substitute the values of y and z back into the first equation \(245x + 145y+ 45z = 485\). Substituting the values of y and z in the equation gives, \(245x + 145(0.191) + 45(3.5) = 485\) Calculating the all the terms and solving for x, we get: \(245x = 377.9\) ⟹ \(x = 1.542\)

Step 6: Final Answer

Therefore, the servings required to meet the overall nutritional requirements are: - 1.542 servings of broiled ground beef (x) - 0.191 servings of baked potato (y) - 3.5 servings of strawberries (z) Note that it might be impractical to serve fractional servings of food items, so it is recommended to round up the servings to the nearest whole number and adjust the menu accordingly.

Key Concepts

Matrix Method

The matrix method is a powerful tool for solving systems of linear equations, especially when there are three or more variables involved. It involves arranging the system of equations into a matrix format, which is a grid of numbers that helps in organizing the coefficients of variables and constants. Here’s how it works in our case:

• We start by writing the coefficients of the variables (x, y, z) from each equation into matrix form. This is called the coefficient matrix.
• Beside it, we write the constants from the equations as a separate column, creating what’s known as the augmented matrix.

In our example, the augmented matrix helps us see the relationships between different servings and nutrients clearly. It serves as a structured way to apply operations that simplify the system and find a solution efficiently.

Gaussian Elimination

Gaussian elimination is a method used to solve linear systems by transforming the augmented matrix into a simpler form. The goal is to make the coefficient matrix into an upper triangular form, which makes it easier to find the values of variables through back substitution.
To perform Gaussian elimination, we do the following:

• We perform row operations such as swapping rows, multiplying a row by a scalar, or adding and subtracting rows.
• The objective is to create zeros below the leading coefficients of each row, simplifying the matrix step-by-step.

In the provided problem, Gaussian elimination transforms the starting matrix into a form where we can directly solve for one variable at a time, starting from the bottom row and working upwards. This systematic reduction is very useful in nutritional planning problems, where precise measures are needed.

Nutritional Planning

Nutritional planning involves creating balanced meals that meet specific dietary requirements, such as calories, carbohydrates, and calcium. In this problem, the dietician must plan a menu using specific servings of beef, potatoes, and strawberries to match nutritional goals.
This problem demonstrates the importance of:

• Understanding nutritional content of each food item.
• Using equations to represent dietary targets, helping to translate them into servings.

Using linear equations for nutritional planning allows us to handle complex requirements efficiently, ensuring that meals provide exactly what is needed without guesswork. It is a practical application of mathematics that ensures dietary guidelines are met in a systematic way.

System of Equations

A system of linear equations includes multiple equations that must be solved together to find a common solution. In this context, the dietician's problem involves three linear equations based on caloric, carbohydrate, and calcium requirements.
Characteristics of systems of equations:

• Each equation represents a different constraint or requirement.
• Solutions are found at the intersection of these equations, representing the point where all requirements are balanced.

For this problem, solving the system gives the makeup of servings needed for different foods, ensuring precise fulfillment of dietary goals. By using methods like the matrix method and Gaussian elimination, these equations are easier to manage, making complex planning tasks achievable.