A health food company is going to make a 600-pound batch of its 'Superfood Blend', which costs $16.35 per pound. This blend consists of whey protein, which costs $10.25 per pound, and powdered spirulina, which costs $22.45 per pound. How many pounds of each should be used to make the blend?

Question

Write your answers as whole numbers or as decimals rounded to the nearest tenth.
pounds of whey protein
pounds of powdered spirulina

Answer 1

Let's define the variables:

• Let \( x \) be the number of pounds of whey protein.
• Let \( y \) be the number of pounds of powdered spirulina.

We have two equations based on the information given:

1. The total weight of the blend must be 600 pounds: \[ x + y = 600 \]

2. The total cost of the blend must equal the cost of making 600 pounds at $16.35 per pound: \[ 10.25x + 22.45y = 16.35 \times 600 \] Calculating the total cost: \[ 16.35 \times 600 = 9810 \] So the second equation is: \[ 10.25x + 22.45y = 9810 \]

Now we can solve this system of equations.

From the first equation, we can express \( y \) in terms of \( x \): \[ y = 600 - x \]

Substituting \( y \) in the second equation: \[ 10.25x + 22.45(600 - x) = 9810 \]

Expanding this: \[ 10.25x + 13470 - 22.45x = 9810 \]

Combining like terms: \[ -12.20x + 13470 = 9810 \]

Subtracting 13470 from both sides: \[ -12.20x = 9810 - 13470 \] \[ -12.20x = -3650 \]

Dividing by -12.20: \[ x = \frac{3650}{12.20} \approx 299.2 \]

Now we can find \( y \): \[ y = 600 - x = 600 - 299.2 \approx 300.8 \]

Thus, rounded to the nearest tenth, we have:

• Pounds of whey protein: \( \approx 299.2 \)
• Pounds of powdered spirulina: \( \approx 300.8 \)

In conclusion, the amounts required are:

• Pounds of whey protein: 299.2
• Pounds of powdered spirulina: 300.8