Question

A dietitian at General Hospital wants a patient to have a meal that has 47 grams (g) of protein, 58 g of carbohydrates, and 630 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is pork chops, corn on the cob, and \(2 \%\) milk. Each serving of pork chops has \(23 \mathrm{~g}\) of protein, \(0 \mathrm{~g}\) of carbohydrates, and \(10 \mathrm{mg}\) of calcium. Each serving of corn on the cob contains \(3 \mathrm{~g}\) of protein, \(16 \mathrm{~g}\) of carbohydrates, and \(10 \mathrm{mg}\) of calcium. Each glass of \(2 \%\) milk contains \(9 \mathrm{~g}\) 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 pork chops, 2 servings of corn on the cob, and 2 glasses of 2% milk.

Step-by-step solution

Define the Variables

Let x be the number of servings of pork chops, y be the number of servings of corn on the cob, and z be the number of glasses of 2% milk. These variables will help set up the system of equations.

Set Up the Protein Equation

Each serving of pork chops has 23 g of protein, each serving of corn on the cob has 3 g of protein, and each glass of 2% milk has 9 g of protein. Set up the equation for total protein: 23x + 3y + 9z = 47

Set Up the Carbohydrates Equation

Each serving of pork chops has 0 g of carbohydrates, each serving of corn on the cob has 16 g of carbohydrates, and each glass of 2% milk has 13 g of carbohydrates. Set up the equation for total carbohydrates: 16y + 13z = 58

Set Up the Calcium Equation

Each serving of pork chops has 10 mg of calcium, each serving of corn on the cob has 10 mg of calcium, and each glass of 2% milk has 300 mg of calcium. Set up the equation for total calcium: 10x + 10y + 300z = 630

Simplify the System of Equations

We have the system of equations:1. 23x + 3y + 9z = 472. 16y + 13z = 583. 10x + 10y + 300z = 630We can simplify the third equation by dividing everything by 10: x + y + 30z = 63

Solve the System of Equations

Using the simplified system:1. 23x + 3y + 9z = 472. 16y + 13z = 583. x + y + 30z = 63First, we solve equation 2 for y: y = (58 - 13z)/16Substitute y in the third equation:x + (58 - 13z)/16 + 30z = 63Solve for x: x = 63 - (58 - 13z)/16 - 30zNext, substitute both y and x into the first equation:23(63 - (58 - 13z)/16 - 30z) + 3((58 - 13z)/16) + 9z = 47Solve for z and then use z to find y and x.

Final Calculation and Verification

Calculate the values of x, y, and z. Verify solutions by substituting them back into the original equations to ensure they meet the nutritional requirements.

Key concepts

System of Equations

A system of equations is a set of multiple equations with multiple variables. In this exercise, the variables represent the servings of different foods (pork chops, corn on the cob, and 2% milk). We aim to find the appropriate number of servings of each food that meet specific nutritional requirements.
To solve a system of equations, we set up mathematical relationships (equations) based on given information. Our goal is to find the values of each variable that satisfy all equations simultaneously. This often involves methods like substitution, elimination, or matrix operations.

Here, we have three main equations based on protein, carbohydrates, and calcium amounts, respectively:
1. Protein: \(23x + 3y + 9z = 47\)
2. Carbohydrates: \(16y + 13z = 58\)
3. Calcium: \(x + y + 30z = 63\) (after simplification).
Solving systems of equations involves steps like isolating variables, substituting values, and verifying solutions.

Nutritional Requirements

Nutritional requirements are the specific amounts of various nutrients needed for maintaining health and proper functioning of the body. In the given exercise, the dietitian must provide a meal that has:

• 47 grams of protein
• 58 grams of carbohydrates
• 630 milligrams of calcium

Each food item in the meal contributes a different amount of these nutrients. Therefore, it's essential to carefully calculate the servings to ensure the patient receives the exact required amounts.

Understanding nutritional requirements is important for diet planning, especially in medical settings where precise nutrient intake can affect recovery and health outcomes significantly.

Diet Planning

Diet planning involves creating meals that meet specific dietary guidelines or nutritional requirements. In this context, the dietitian at General Hospital plans a meal with pork chops, corn on the cob, and 2% milk to meet the patient's specific nutrient needs.

Effective diet planning requires a deep understanding of the nutritional content of food items and how they can be combined to meet dietary goals. Here, we calculate servings of each food to balance the patient's diet:

• Pork chops (high in protein)
• Corn on the cob (carbohydrates and supplemental protein)
• 2% milk (good source of calcium and protein)

In practice, diet planning is a crucial skill for dietitians. They ensure that meals are not only nutritious but also delicious and enjoyable for the patients. Proper diet planning can significantly impact health, recovery, and overall well-being.