A glass of skim milk supplies 0.1 mg of iron and 8.4 g of protein. A quarter pound of lean meat provides 3.2 mg of iron and 22 g of protein. A person on a special diet is to have 13.3 mg of iron and 130.0 g of protein. How many glasses of skim milk and how many​ quarter-pound servings of meat will provide​ this?

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Part 1
How many​ glass(es) of skim milk and how many​ quarter-pound serving(s) of lean meat would be​ required?
  
enter your response here ​glass(es) of skim milk and
  
enter your response here ​quarter-pound serving(s) of lean meat will provide the needed nutrients.

Answer 1

Let \( x \) be the number of glasses of skim milk and \( y \) be the number of quarter-pound servings of meat. We can set up the following system of equations based on the nutrient contributions from each food item.

Nutrient Information:

• Skim milk:
  • Iron: \( 0.1 \) mg per glass
  • Protein: \( 8.4 \) g per glass
• Lean meat:
  • Iron: \( 3.2 \) mg per quarter-pound serving
  • Protein: \( 22 \) g per quarter-pound serving

Formulating the Equations:

We need to satisfy the following two conditions:

1. For iron: \( 0.1x + 3.2y = 13.3 \)
2. For protein: \( 8.4x + 22y = 130.0 \)

Solve the System of Equations:

1. From the Iron Equation: \[ 0.1x + 3.2y = 13.3 \tag{1} \]

2. From the Protein Equation: \[ 8.4x + 22y = 130.0 \tag{2} \]

We can solve Equation (1) for \( x \): \[ 0.1x = 13.3 - 3.2y \] \[ x = \frac{13.3 - 3.2y}{0.1} \] \[ x = 133 - 32y \tag{3} \]

Now, substitute Equation (3) into Equation (2): \[ 8.4(133 - 32y) + 22y = 130 \] \[ 1117.2 - 268.8y + 22y = 130 \] Combine the \( y \) terms: \[ 1117.2 - 246.8y = 130 \] Rearranging gives: \[ -246.8y = 130 - 1117.2 \] \[ -246.8y = -987.2 \] Dividing by -246.8 gives: \[ y = \frac{987.2}{246.8} \approx 4 \]

Now substitute \( y \) back into Equation (3) to find \( x \): \[ x = 133 - 32(4) \] \[ x = 133 - 128 = 5 \]

Final Answer:

• Glasses of skim milk: \( 5 \)
• Quarter-pound servings of meat: \( 4 \)

Thus, the answer is: 5 glasses of skim milk and 4 quarter-pound servings of lean meat will provide the needed nutrients.