Math Problem Statement

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?

Solution

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 xx be the number of tubes of food A.
  • Let yy 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=424x + 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=482x + 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=162x + y = 16

Now we have a system of three equations:

  1. 4x+3y=424x + 3y = 42
  2. 2x+6y=482x + 6y = 48
  3. 2x+y=162x + y = 16

Solving the system of equations:

Step 1: Solve the third equation for yy:

From 2x+y=162x + y = 16, we can solve for yy: y=162xy = 16 - 2x

Step 2: Substitute y=162xy = 16 - 2x into the first equation:

Substituting into 4x+3y=424x + 3y = 42: 4x+3(162x)=424x + 3(16 - 2x) = 42 Expand and solve for xx: 4x+486x=424x + 48 - 6x = 42 2x+48=42-2x + 48 = 42 2x=6-2x = -6 x=3x = 3

Step 3: Substitute x=3x = 3 into y=162xy = 16 - 2x:

y=162(3)=166=10y = 16 - 2(3) = 16 - 6 = 10

Final Answer:

  • The number of tubes of food A is x=3x = 3.
  • The number of tubes of food B is y=10y = 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Substitution Method

Formulas

4x + 3y = 42 (Protein requirement)
2x + 6y = 48 (Carbohydrates requirement)
2x + y = 16 (Fat requirement)

Theorems

System of Linear Equations
Substitution and Elimination

Suitable Grade Level

Grades 9-12