Math Problem Statement

A dietitian is planning a diet that must include exactly 300 grams of protein, 400 grams of carbohydrates, and 200 grams of fat. She has three food sources to choose from, each with the following nutrient composition per unit: Food A: 10 grams of protein, 20 grams of carbohydrates, 5 grams of fat. Food B: 20 grams of protein, 10 grams of carbohydrates, 15 grams of fat. Food C: 30 grams of protein, 40 grams of carbohydrates, 10 grams of fat. How many units of each food source should she include in the diet to meet the exact nutrient requirements? (Use Gauss Jordan)

Solution

The problem asks to find how many units of each food (A, B, C) a dietitian should include to meet the exact nutrient requirements: 300 grams of protein, 400 grams of carbohydrates, and 200 grams of fat, using the Gauss-Jordan method.

Step 1: Define variables

Let:

  • x1x_1 be the number of units of Food A,
  • x2x_2 be the number of units of Food B,
  • x3x_3 be the number of units of Food C.

Step 2: Set up the system of equations

From the given nutrient composition per unit:

  1. Protein equation:
    10x1+20x2+30x3=30010x_1 + 20x_2 + 30x_3 = 300 (since we need 300 grams of protein).

  2. Carbohydrate equation:
    20x1+10x2+40x3=40020x_1 + 10x_2 + 40x_3 = 400 (since we need 400 grams of carbohydrates).

  3. Fat equation:
    5x1+15x2+10x3=2005x_1 + 15x_2 + 10x_3 = 200 (since we need 200 grams of fat).

This leads to the following system of equations:

10x_1 + 20x_2 + 30x_3 &= 300 \\ 20x_1 + 10x_2 + 40x_3 &= 400 \\ 5x_1 + 15x_2 + 10x_3 &= 200 \end{align*}$$ ### Step 3: Gauss-Jordan Elimination We can now solve this system using the Gauss-Jordan method. I’ll perform the necessary row operations to find the solution.The solution to the system using Gauss-Jordan elimination gives the following values for the number of units of each food: - $$x_1 = 20$$ units of Food A, - $$x_2 = 8$$ units of Food B, - $$x_3 = -2$$ units of Food C. Since the result suggests a negative number of units for Food C, this solution implies that it is not feasible to use Food C in this diet. A better approach might be to recheck the constraints or adjust the problem setup, possibly avoiding negative units. Would you like a detailed explanation of the Gauss-Jordan steps or further clarification? Here are some related questions for practice: 1. What happens if you remove Food C from the diet plan? 2. How would you adjust the system if the diet included more than three types of food? 3. Can you solve the system using a different method, such as matrix inversion? 4. What modifications to nutrient requirements might lead to non-negative solutions for all foods? 5. How does changing the nutrient composition of one of the foods affect the system of equations? **Tip**: Always verify the practicality of solutions in real-life contexts, like ensuring no negative values when calculating quantities!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Matrix Operations

Formulas

Protein equation: 10x_1 + 20x_2 + 30x_3 = 300
Carbohydrate equation: 20x_1 + 10x_2 + 40x_3 = 400
Fat equation: 5x_1 + 15x_2 + 10x_3 = 200

Theorems

Gauss-Jordan Elimination

Suitable Grade Level

College/Advanced High School

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