After taking a nutrition​ class, a big macaroni and cheese fan decides to improve the levels of protein and fiber in her favorite lunch by adding broccoli and canned chicken. The nutritional information for the foods referred to in this exercise is given in the table below. Complete parts​ (a) and​ (b) below.
Nutrition Information per Serving
Nutrient
Mac and Cheese
Broccoli
Chicken
Shells
Calories
269
51
69
259
Protein​ (g)
10
5.3
15
9
Fiber​ (g)
1
5.2
0
4
Question content area bottom
Part 1
a. If she wants to limit her lunch to 390 calories but get 35 grams of protein and 10 grams of​ fiber, what proportions of servings of macaroni and​ cheese, broccoli, and chicken should she​ use?
She should use 
  
enter your response here servings of macaroni and​ cheese, 
  
enter your response here servings of​ broccoli, and 
  
enter your response here servings of chicken to get her desired nutritional content.
​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)

Let's define the variables for each food item:

- Let \( x \) be the servings of macaroni and cheese.
- Let \( y \) be the servings of broccoli.
- Let \( z \) be the servings of chicken.

We are given the following nutritional information per serving for each food item:

| Nutrient       | Mac & Cheese (M) | Broccoli (B) | Chicken (C) |
|----------------|------------------|--------------|-------------|
| Calories       | 269              | 51           | 69          |
| Protein (g)    | 10               | 5.3          | 15          |
| Fiber (g)      | 1                | 5.2          | 0           |

### The goal is to solve for \( x \), \( y \), and \( z \) given these constraints:
1. **Calories:** Total calories should be 390.
   \[
   269x + 51y + 69z = 390
   \]

2. **Protein:** Total protein should be 35 grams.
   \[
   10x + 5.3y + 15z = 35
   \]

3. **Fiber:** Total fiber should be 10 grams.
   \[
   1x + 5.2y + 0z = 10
   \]

We now have a system of three equations:
\[
\begin{aligned}
1. & \ 269x + 51y + 69z = 390 \\
2. & \ 10x + 5.3y + 15z = 35 \\
3. & \ x + 5.2y = 10
\end{aligned}
\]

### Step 1: Solve the third equation for \( x \):
\[
x = 10 - 5.2y
\]

### Step 2: Substitute \( x = 10 - 5.2y \) into the other two equations.

#### Substitute into the first equation (calories):
\[
269(10 - 5.2y) + 51y + 69z = 390
\]
Simplifying:
\[
2690 - 1398.8y + 51y + 69z = 390
\]
\[
2690 - 1347.8y + 69z = 390
\]
\[
-1347.8y + 69z = -2300
\]
\[
1347.8y - 69z = 2300 \quad \text{(Equation A)}
\]

#### Substitute into the second equation (protein):
\[
10(10 - 5.2y) + 5.3y + 15z = 35
\]
Simplifying:
\[
100 - 52y + 5.3y + 15z = 35
\]
\[
100 - 46.7y + 15z = 35
\]
\[
-46.7y + 15z = -65
\]
\[
46.7y - 15z = 65 \quad \text{(Equation B)}
\]

### Step 3: Solve the system of two equations (Equation A and Equation B).

#### Equation A:
\[
1347.8y - 69z = 2300
\]

#### Equation B:
\[
46.7y - 15z = 65
\]

I will solve these two equations now.The solution to the system of equations is:

- \( y \approx 1.77 \) (servings of broccoli)
- \( z \approx 1.17 \) (servings of chicken)

### Step 4: Substitute these values into \( x = 10 - 5.2y \) to find \( x \) (servings of macaroni and cheese):
\[
x = 10 - 5.2 \times 1.77
\]

Let me calculate that value for \( x \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate \( x \) using the formula:
\[
x = 10 - 5.2 \times 1.77
\]

This should give you the amount of macaroni and cheese servings. Let me know if you'd like more assistance or further explanation!

### Related questions:
1. What happens to the nutritional content if she increases the calorie limit?
2. How would the proportions change if she wanted to increase her fiber intake to 15 grams?
3. What is the overall fiber increase compared to just having macaroni and cheese?
4. How many servings of each would provide exactly 400 calories?
5. What if she only wanted 30 grams of protein but kept the same calorie and fiber goals?

**Tip:** To make similar calculations, try breaking the problem into smaller steps and solving equations sequentially!

After taking a nutrition class, a big macaroni and cheese fan decides to improve the levels of protein and fiber in her favorite lunch by adding broccoli and canned chicken. The nutritional information for the foods referred to in this exercise is given in the table below. If she wants to limit her lunch to 390 calories but get 35 grams of protein and 10 grams of fiber, what proportions of servings of macaroni and cheese, broccoli, and chicken should she use?

Discover how to solve a nutritional optimization problem using linear equations. This example demonstrates the application of a system of linear equations to determine the optimal proportions of macaroni and cheese, broccoli, and chicken to meet specific dietary goals. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation