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 280 grams of carbohydrates, 200 grams of fat, and 140 grams of protein. He knows that each cup of rice provides 48.6 grams of carbohydrates and 2 grams of protein, but no fat at all.  Each cup of tofu provides  6.6 grams of carbohydrates, 11 grams of fat, and 26 grams of protein. Finally, each cup of peanuts provides  34 grams of carbohydrates, 65.5 grams of fat, and 32 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:   

cups of tofu:   

cups of peanuts:   

To solve this problem, let's set up the system of equations based on the given nutritional information.

### Variables:
- Let \( x \) represent the number of cups of rice.
- Let \( y \) represent the number of cups of tofu.
- Let \( z \) represent the number of cups of peanuts.

### Equations based on nutritional goals:

1. **Carbohydrates**: 
   \[
   48.6x + 6.6y + 34z = 280
   \]
2. **Fat**: 
   \[
   0x + 11y + 65.5z = 200
   \]
3. **Protein**: 
   \[
   2x + 26y + 32z = 140
   \]

We can now solve this system of equations using a matrix method. I'll calculate the solution.The solution to the system of equations is:

- Cups of rice: \( 3.594 \)
- Cups of tofu: \( 1.702 \)
- Cups of peanuts: \( 2.768 \)

These amounts of rice, tofu, and peanuts will meet Juan's nutritional goals of 280 grams of carbohydrates, 200 grams of fat, and 140 grams of protein.

Would you like a breakdown of how these numbers satisfy the nutritional requirements? 

Here are 5 related questions to consider:
1. How would the nutritional content change if Juan wanted to reduce his fat intake to 150 grams?
2. What would happen if Juan added a fourth food item to his meal, such as vegetables with no fat but extra protein?
3. How would the solution change if we considered different measurement units, such as grams instead of cups?
4. Could Juan achieve the same nutritional goal using different proportions of tofu and peanuts?
5. What happens if the nutritional goals are increased to meet a higher-calorie requirement?

**Tip:** Solving nutritional problems using matrices helps streamline complex systems of equations with multiple variables.

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 280 grams of carbohydrates, 200 grams of fat, and 140 grams of protein. He knows that each cup of rice provides 48.6 grams of carbohydrates and 2 grams of protein, but no fat at all. Each cup of tofu provides 6.6 grams of carbohydrates, 11 grams of fat, and 26 grams of protein. Finally, each cup of peanuts provides 34 grams of carbohydrates, 65.5 grams of fat, and 32 grams of protein. How many cups of rice, tofu, and peanuts should he eat?

Learn how to solve a nutritional planning problem using linear algebra concepts such as matrices and systems of linear equations. This problem involves calculating the number of cups of rice, tofu, and peanuts needed to meet specific nutritional goals for carbohydrates, fat, and protein. Suitable for students in Grades 10-12.

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