A​ nutritionist, working for​ NASA, must meet certain minimum nutritional requirements and yet keep the weight of the food at a minimum. He is considering a combination of two​ foods, which are packaged in tubes. Each tube of food A contains 4 units of​ protein, 2 units of​ carbohydrates, and 2 units of fat and weighs 2 pounds. Each tube of food B contains 3 units of​ protein, 6 units of​ carbohydrates, and 1 unit of fat and weighs 2 pounds. The requirement calls for 42 units of​ protein, 48 units of​ carbohydrates, and 16 units of fat. How many tubes of each food should be supplied to the​ astronauts?

This problem can be solved using a system of linear equations. Let's define variables based on the number of tubes of each type of food:

- Let \( x \) be the number of tubes of food A.
- Let \( y \) be the number of tubes of food B.

The nutritional and weight requirements for the astronauts are:

### Protein Requirement:
- Each tube of food A provides 4 units of protein.
- Each tube of food B provides 3 units of protein.
- The total required protein is 42 units.

This gives the equation:
\[
4x + 3y = 42
\]

### Carbohydrates Requirement:
- Each tube of food A provides 2 units of carbohydrates.
- Each tube of food B provides 6 units of carbohydrates.
- The total required carbohydrates are 48 units.

This gives the equation:
\[
2x + 6y = 48
\]

### Fat Requirement:
- Each tube of food A provides 2 units of fat.
- Each tube of food B provides 1 unit of fat.
- The total required fat is 16 units.

This gives the equation:
\[
2x + y = 16
\]

Now we have a system of three equations:
1. \( 4x + 3y = 42 \)
2. \( 2x + 6y = 48 \)
3. \( 2x + y = 16 \)

### Solving the system of equations:

#### Step 1: Solve the third equation for \( y \):
From \( 2x + y = 16 \), we can solve for \( y \):
\[
y = 16 - 2x
\]

#### Step 2: Substitute \( y = 16 - 2x \) into the first equation:
Substituting into \( 4x + 3y = 42 \):
\[
4x + 3(16 - 2x) = 42
\]
Expand and solve for \( x \):
\[
4x + 48 - 6x = 42
\]
\[
-2x + 48 = 42
\]
\[
-2x = -6
\]
\[
x = 3
\]

#### Step 3: Substitute \( x = 3 \) into \( y = 16 - 2x \):
\[
y = 16 - 2(3) = 16 - 6 = 10
\]

### Final Answer:
- The number of tubes of food A is \( x = 3 \).
- The number of tubes of food B is \( y = 10 \).

Thus, the astronauts should be supplied with 3 tubes of food A and 10 tubes of food B.

Would you like further details or have any questions?

---

Here are 5 related questions to expand on this problem:
1. How does the total weight of the food change with different combinations of food A and food B?
2. How would the solution change if the nutritional requirements were increased by 20%?
3. What would happen if food A had a different weight, say 3 pounds per tube?
4. How could you solve this problem graphically using the system of equations?
5. Can this problem be modeled using a linear programming approach to minimize weight?

**Tip:** When solving systems of equations, substitution and elimination methods are helpful strategies to simplify the problem step-by-step.

A nutritionist, working for NASA, must meet certain minimum nutritional requirements and yet keep the weight of the food at a minimum. He is considering a combination of two foods, which are packaged in tubes. Each tube of food A contains 4 units of protein, 2 units of carbohydrates, and 2 units of fat and weighs 2 pounds. Each tube of food B contains 3 units of protein, 6 units of carbohydrates, and 1 unit of fat and weighs 2 pounds. The requirement calls for 42 units of protein, 48 units of carbohydrates, and 16 units of fat. How many tubes of each food should be supplied to the astronauts?

Learn how to solve a system of linear equations to meet specific nutritional requirements for astronauts. This problem involves calculating the correct combination of two foods to satisfy protein, carbohydrates, and fat needs while minimizing weight. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation