Math Problem Statement

Juan is trying to plan a meal to meet specific nutritional goals. He wants to prepare a meal containing rice, tofu, and peanuts that will provide 220 grams of carbohydrates, 200 grams of fat, and 140 grams of protein. He knows that each cup of rice provides 43.4 grams of carbohydrates and 3 grams of protein, but no fat at all. Each cup of tofu provides 6.8 grams of carbohydrates, 10 grams of fat, and 25.4 grams of protein. Finally, each cup of peanuts provides 33.5 grams of carbohydrates, 65.5 grams of fat, and 29 grams of protein. How many cups of rice, tofu, and peanuts should he eat?

Write a system of equations and then solve using the calculator and a matrix. Do not solve by hand. {If necessary, give at least 3 decimal places.}

cups of rice: Incorrect

cups of tofu:

cups of peanuts:

Solution

To solve this problem, we need to set up a system of equations based on the nutritional information provided for rice, tofu, and peanuts, and Juan's goals of reaching 220 grams of carbohydrates, 200 grams of fat, and 140 grams of protein. Let’s define the variables:

  • xx: cups of rice
  • yy: cups of tofu
  • zz: cups of peanuts

Based on the nutritional dataTo solve this problem, we need to set up a system of equations based on the nutritional information provided for rice, tofu, and peanuts, and Juan's goals of reaching 220 grams of carbohydrates, 200 grams of fat, and 140 grams of protein. Let’s define the variables:

  • xx: cups of rice
  • yy: cups of tofu
  • zz: cups of peanuts

Based on the nutritional

  • Carbohydrates:

    • Rice: 43.4 grams per cup
    • Tofu: 6.8 grams per cup
    • Peanuts: 33.5 grams per cup

  • Fat:

    • Rice: 0 grams per cup
    • Tofu: 10 grams per cup
    • Peanuts: 65.5 grams per cup

  • Protein:

    • Rice: 3 grams per cup
    • Tofu: 25.4 grams per cup
    • Peanuts: 29 grams per cup

The system of equations becomes:

  1. Carbohydrates:
    43.4x+6.8y+33.5z=22043.4x + 6.8y + 33.5z = 220

  2. Fat:
    0x+10y+65.5z=2000x + 10y + 65.5z = 200

  3. Protein:
    3x+25.4y+29z=1403x + 25.4y + 29z = 140

Now we can solve this system using matrix methods. I will calculate the solution using these equations.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can help walk you through the process of solving it step by step manually or suggest another approach.

Would you like more details or have any other questions?

Here are 5 related questions:

  1. How would you solve a system of equations manually using substitution?
  2. Can you express the equations graphically and interpret their intersection?
  3. What other food options could meet the same nutritional requirements?
  4. How do matrix methods apply to real-world nutrition problems?
  5. What role does linear algebra play in optimizing diet plans?

Tip: When solving systems of equations, always verify the units (e.g., cups, grams) to ensure consistency across all variables.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
System of Equations
Matrices

Formulas

Carbohydrate equation: 43.4x + 6.8y + 33.5z = 220
Fat equation: 10y + 65.5z = 200
Protein equation: 3x + 25.4y + 29z = 140

Theorems

Matrix Inversion
Gaussian Elimination

Suitable Grade Level

College Level / Advanced High School (Grade 11-12)

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