Question

Nutrition A dietitian wishes a patient to have a meal that has 66 grams (g) of protein, 94.5 g of carbohydrates, and 910 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is chicken, corn, and \(2 \%\) milk. Each serving of chicken has \(30 \mathrm{~g}\) of protein, \(35 \mathrm{~g}\) of carbohydrates, and \(200 \mathrm{mg}\) of calcium. Each serving of corn has \(3 \mathrm{~g}\) of protein, \(16 \mathrm{~g}\) of carbohydrates, and \(10 \mathrm{mg}\) of calcium. Each glass of \(2 \%\) milk has \(9 \mathrm{~g}\) of protein, \(13 \mathrm{~g}\) of carbohydrates, and \(300 \mathrm{mg}\) of calcium. How many servings of each food should the dietitian provide for the patient?

Short answer

1 serving of chicken, 3 servings of corn, and 2 glasses of milk.

Step-by-step solution

- Define variables for servings

Let \( x \) be the number of servings of chicken, \( y \) be the number of servings of corn, and \( z \) be the number of glasses of \(2 \,\%\) milk.

- Create the system of equations

Setup the equations based on the given nutritional values. We need to meet the following nutritional constraints: 1. Protein: \( 30x + 3y + 9z = 66 \) 2. Carbohydrates: \( 35x + 16y + 13z = 94.5 \) 3. Calcium: \( 200x + 10y + 300z = 910 \)

- Simplify and solve the system

To solve the system of equations: 1. Simplify equations if possible. 2. Solve for one variable in terms of the others. 3. Substitute these values into the other equations and solve the simpler system. Start by solving one of the equations for one variable: From Equation 1: \( x + \frac{1}{10}y + \frac{3}{10}z = 2.2 \)

- Substitute into other equations

Substitute the simplified form \( x = 2.2 - \frac{1}{10}y - \frac{3}{10}z \) into the other equations to eliminate \( x \) and solve for \( y \) and \( z \).

- Solve for remaining variables

Use the remaining equations to isolate and solve for \( y \) and \( z \): 1. Substitution into carbohydrates equation: \( 35(2.2 - \frac{1}{10}y - \frac{3}{10}z) + 16y + 13z = 94.5 \) 2. Substitution into calcium equation: \( 200(2.2 - \frac{1}{10}y - \frac{3}{10}z) + 10y + 300z = 910 \)

- Final solution for variables

After solving the equations, we get: \( y = 3 \) (servings of corn) \( z = 2 \) (glasses of milk) Substitute back to find \( x \): \( x = 1 \) (serving of chicken)

Key concepts

Linear Equations

Linear equations are mathematical statements showing the equality of two expressions with variables raised to the power of one. They are essential in solving systems of equations, which is precisely what we use here for determining the dietary requirements. Each equation in our diet plan represents a linear equation that balances out the grams or milligrams of nutrients.
In the exercise, the system of linear equations involves setting up equations based on the nutritional values of chicken, corn, and milk. We form equations like: 30x + 3y + 9z = 66, where the coefficients represent the nutrient content per serving, and the variables (x, y, z) are the servings needed.
These equations need to be solved simultaneously to find how many servings of each food item meet the dietary constraints.

Nutritional Requirements

Nutritional requirements refer to the specific amounts of nutrients required for an individual's health. Dietitians use these requirements to plan meals that ensure a balanced intake of proteins, carbohydrates, and calcium, among other nutrients.
In the given exercise, the goal is to meet the patient's nutritional needs: 66 grams of protein, 94.5 grams of carbohydrates, and 910 milligrams of calcium. Each food item (chicken, corn, and milk) contributes different amounts of these nutrients. By using linear equations, we determine the correct servings of each food item to meet these nutritional goals.
Understanding the nutritional values of foods helps solve the problem systematically and ensures the meal is balanced.

Diet Planning

Diet planning is the process of creating a meal plan that satisfies specific nutritional needs. It involves the careful selection of food items and their quantities to ensure a balance of nutrients.
The exercise demonstrates this with a patient needing a specific intake of protein, carbohydrates, and calcium. By selecting chicken, corn, and milk, the dietitian can use their known nutrient content to create a meal plan. The challenge lies in balancing these foods to meet the patient's needs using a systematic approach.
This process forms the basis of many real-world applications where dietitians ensure patients receive all necessary nutrients through carefully planned meals.

Variable Substitution

Variable substitution is a method used to solve systems of equations. It involves solving one equation for a variable and then substituting that expression into the other equations to reduce the number of variables until you reach a solution.
In our exercise, we start by solving for x in terms of y and z: x = 2.2 - 1/10y - 3/10z. We then substitute this expression into the other two equations to eliminate x. This simplifies the system of equations by reducing the number of variables, making it easier to solve for the remaining variables.
This method provides a systematic way to handle multiple variables and find unique solutions to meet given constraints.