Math Problem Statement

A dietitian wishes to plan a meal around three foods. The meal is to include 7,600 units of vitamin A, 2,810 units of vitamin C, and 990 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the following table. Food I Food II Food III Vitamin A 400 1200 800 Vitamin C 110   570   340
Calcium 90   30   60

Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements. (Let x, y, and z denote the number of ounces of Food I, Food II, and Food III, respectively. If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, express your answer in terms of the parameters t and/or s.) (x, y, z) =

Solution

We need to determine how many ounces of each food should be included in the meal such that the vitamin A, vitamin C, and calcium requirements are satisfied. The number of units of each nutrient in each food is summarized in the table.

Let:

  • xx be the number of ounces of Food I,
  • yy be the number of ounces of Food II,
  • zz be the number of ounces of Food III.

We are given the following system of equations based on the requirements of the nutrients:

  1. Vitamin A Requirement: 400x+1200y+800z=7600400x + 1200y + 800z = 7600

  2. Vitamin C Requirement: 110x+570y+340z=2810110x + 570y + 340z = 2810

  3. Calcium Requirement: 90x+30y+60z=99090x + 30y + 60z = 990

We can solve this system of equations to find xx, yy, and zz.

Step 1: Solve the system of equations

Let me calculate the solution.The system of equations has infinitely many solutions, which can be expressed in terms of the parameter zz:

x=10z2x = 10 - \frac{z}{2} y=3z2y = 3 - \frac{z}{2}

This means the dietitian can choose any value for zz, and the corresponding values of xx and yy will satisfy the vitamin and calcium requirements.

Thus, the solution is:

(x,y,z)=(10z2,3z2,z)(x, y, z) = \left( 10 - \frac{z}{2}, 3 - \frac{z}{2}, z \right)

If you'd like further details on how to interpret this solution, feel free to ask!

Follow-up questions:

  1. What would be the values of xx, yy, and zz if z=2z = 2?
  2. Can the dietitian choose a negative value for zz?
  3. How does each food contribute to the nutrient intake?
  4. What is the relationship between zz and the ounces of food needed?
  5. How does the choice of zz impact the total amount of food required?

Tip:

When working with infinitely many solutions, always check if the parameters (like zz) make sense in the real-world context. For instance, negative quantities of food are usually not feasible.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Parametric Solutions

Formulas

400x + 1200y + 800z = 7600
110x + 570y + 340z = 2810
90x + 30y + 60z = 990

Theorems

Solution of Systems of Linear Equations
Parametric Solution of Infinite Systems

Suitable Grade Level

Grades 10-12