Problem 19 - Elementary and Intermediate Algebra

Question

Nutrition. A dietician in a hospital prepares meals under the guidance of a physician. Suppose that for a particular patient a physician prescribes a meal to have 800 calories, \(55 \mathrm{~g}\) of protein, and \(220 \mathrm{mg}\) of vitamin C. The dietician prepares a meal of roast beef, baked potatoes, and broccoli according to the data in the table below. $$ \begin{array}{|l|c|c|c|} \hline \begin{array}{c} \text { Serving } \\ \text { Size } \end{array} & \text { Calories } & \begin{array}{c} \text { Protein } \\ \text { (in grams) } \end{array} & \begin{array}{c} \text { Vitamin C } \\ \text { (in milligrams) } \end{array} \\ \hline \begin{array}{l} \text { Roast } \\ \text { Beef, } 3 \text { oz } \end{array} & 300 & 20 & 0 \\ \hline \begin{array}{l} \text { Baked } \\ \text { Potato, } 1 \end{array} & 100 & 5 & 20 \\ \hline \begin{array}{l} \text { Broccoli, } \\ 156 \mathrm{~g} \end{array} & 50 & 5 & 100 \\ \hline \end{array} $$ How many servings of each food are needed in order to satisfy the doctor's orders?

Step-by-Step Solution

Answer

2 servings of roast beef, 1 serving of baked potato, and 2 servings of broccoli.

Step 1: Set Up Variables

Let \(x\) be the number of servings of roast beef, \(y\) be the number of servings of baked potatoes, and \(z\) be the number of servings of broccoli. We need to find the values of \(x\), \(y\), and \(z\) that meet the nutritional requirements.

Step 2: Create System of Equations

From the problem, set up the following equations based on calories, protein, and vitamin C: For calories: \[ 300x + 100y + 50z = 800 \]For protein: \[ 20x + 5y + 5z = 55 \]For vitamin C: \[ 0x + 20y + 100z = 220 \]

Step 3: Simplify the Equations

Simplify each of the equations if possible: Equation 1 (Calories): \[ 300x + 100y + 50z = 800 \]Equation 2 (Protein): \[ 20x + 5y + 5z = 55 \] or simplified \[ 4x + y + z = 11 \]Equation 3 (Vitamin C): \[ 20y + 100z = 220 \] or simplified \[ y + 5z = 11 \]

Step 4: Solve for Variables

Solve Equation 3 for \(y\): \[ y + 5z = 11 \] \[ y = 11 - 5z \] Substitute \(y = 11 - 5z\) into Equation 2: \[ 4x + (11 - 5z) + z = 11 \] \[ 4x + 11 - 4z = 11 \] \[ 4x - 4z = 0 \] \[ x = z \]

Step 5: Substitute Back to Find Remaining Variables

Use \(x = z\) and substitute into the original equations. From Equation 1 (Calories): \[ 300x + 100(11-5z) + 50z = 800 \]\[ 300z + 100(11 - 5z) + 50z = 800 \]\[ 300z + 1100 - 500z + 50z = 800 \]\[ -150z + 1100 = 800 \]\[ -150z = -300 \]\[ z = 2 \]Then \(x = z = 2\)Substitute back to find \(y\): \[ y = 11 - 5(2) \]\[ y = 1 \]

Step 6: Verify the Solution

Check that the values satisfy all the nutritional requirements:Calories: \[ 300(2) + 100(1) + 50(2) = 600 + 100 + 100 = 800 \]Protein: \[ 20(2) + 5(1) + 5(2) = 40 + 5 + 10 = 55 \]Vitamin C: \[ 20(1) + 100(2) = 20 + 200 = 220 \]The solution is valid.

Key Concepts

Variables

In problems that involve multiple values to be determined, we use variables. Here, the dietician needs to determine the number of servings of roast beef, baked potatoes, and broccoli. We use:

• x for servings of roast beef,
• y for servings of baked potatoes,
• z for servings of broccoli.

By defining these variables, we can set up equations to solve the problem. This approach avoids confusion and ensures our equations accurately represent the nutritional requirements.

Calories

Calories are a measure of energy intake from food. Here, the physician prescribed a meal of 800 calories. Each food item has a specific caloric content per serving. For example, roast beef has 300 calories per 3 oz serving, baked potatoes have 100 calories per serving, and broccoli has 50 calories per 156g. The total calories equation from the given problem is:
i.e., 300x + 100y + 50z = 800.This equation combines the caloric contributions of all servings, which helps ensure our meal meets the required energy intake.

Protein Intake

Protein is essential for body function and muscle repair. The prescribed meal must contain 55 grams of protein. Each food item again contributes differently. Roast beef offers 20 grams per 3 oz, baked potatoes 5 grams per serving, and broccoli also 5 grams per 156g. Thus, for protein, we set up the equation:
20x + 5y + 5z = 55.
This can be simplified to:
4x + y + z = 11.
This way, we account for total protein intake through the servings.

Vitamin C

Vitamin C is vital for immune function. Here, our meal must contain 220mg of Vitamin C. Unlike calories and protein, only baked potatoes and broccoli contribute to Vitamin C. Roast beef does not contribute any Vitamin C. The equation simplifies to:
20y + 100z = 220.
Or simplified further:
y + 5z = 11.
This equation ensures that the combination of servings meets the required Vitamin C intake.

Simplification

Before solving, we make our equations easier to manage by simplifying them where possible. This involves combining like terms or isolating one variable. For instance, the protein equation simplifies from 20x + 5y + 5z = 55 to 4x + y + z = 11 and the Vitamin C equation simplifies from 20y + 100z = 220 to y + 5z = 11. Simplifying helps make the calculations more straightforward and less prone to errors.

Substitution

Substitution is a key method in solving systems of equations. It involves solving one equation for one variable and then replacing that expression in other equations. Here, we solve y + 5z = 11 to get y = 11 - 5z and substitute this into the protein equation 4x + y + z = 11 to eventually find the values of x, y, and z. This method provides a systematic way to find the solution for multiple variables.