Problem 79 - Precalculus

Question

A nutritionist wishes to market a new vitamin-enriched fruit drink and is preparing two versions of it to distribute at a local health club. She has 100 cups of pineapple juice and 4 cups of super vitamin-enriched pomegranate concentrate. One version of the drink is to contain 2\% pomegranate and the other version 4\% pomegranate. How much of each drink can she create if drinks are 1 cup and she uses all of the ingredients?

Step-by-Step Solution

Answer

0 cups of 2% drink and 100 cups of 4% drink.

Step 1: Define Variables

Let's define: - Let \( x \) be the number of 2% pomegranate drinks she can make. - Let \( y \) be the number of 4% pomegranate drinks she can make.

Step 2: Set Up the Equations

We have two main constraints based on the ingredients availability: 1. The total number of cups of pineapple juice used: \[ x + y = 100 \] 2. The total number of cups of pomegranate used: \[ 0.02x + 0.04y = 4 \]

Step 3: Solve Equation for One Variable

From the first equation, express \( x \) in terms of \( y \): \[ x = 100 - y \]

Step 4: Substitute and Solve

Substitute \( x = 100 - y \) into the second equation: \[ 0.02(100-y) + 0.04y = 4 \] Simplify and solve the equation:\[ 2 - 0.02y + 0.04y = 4 \] \[ 0.02y = 2 \] \[ y = 100 \]

Step 5: Find the Value of the Second Variable

Now, substitute \( y = 100 \) back into the equation for \( x \): \[ x = 100 - 100 = 0 \]

Step 6: Verify the Solution

Check if these values satisfy the original conditions:- The total cups: \( x + y = 100 + 0 = 100 \) (Yes) - The total pomegranate concentrate: \( 0.02 \times 0 + 0.04 \times 100 = 4 \) (Yes)

Step 7: Conclusion

She can create 0 cups of the 2% pomegranate drink and 100 cups of the 4% pomegranate drink using all the ingredients.

Key Concepts

Systems of Equations

Systems of equations involve solving two or more equations that share the same variables. They allow us to find values for these variables, which solve all equations simultaneously. In this scenario, the nutritionist is dealing with two types of drinks needing specific amounts of ingredients. By setting up equations based on the constraints of the available pineapple juice and pomegranate concentrate, we represent the problem mathematically.

• The first equation, \( x + y = 100 \), ensures that all 100 cups of pineapple juice are used.
• The second equation, \( 0.02x + 0.04y = 4 \), dictates the exact use of the 4 cups of pomegranate concentrate.

These equations form the system that we must solve to find out how many cups of each type of drink can be made.

Substitution Method

The substitution method is a technique used in algebra to solve systems of equations by expressing one variable in terms of another. Essentially, you isolate one variable and then substitute it into the other equation.

For our nutritional drink scenario, we solve the first equation \( x + y = 100 \) for \( x \), giving us \( x = 100 - y \). This provides a direct way to express \( x \) using \( y \).Next, you substitute \( x = 100 - y \) into the second equation:

• Replace \( x \) in \( 0.02x + 0.04y = 4 \) with \( 100 - y \).
• Simplify this equation to solve for \( y \).

This step reveals that \( y = 100 \), telling us the maximum number of 4% drinks the nutritionist can make. This method simplifies solving complex problems by reducing them to one unknown that we can easily compute.

Solution Verification

Once you have a solution, you should always verify it to ensure accuracy. Solution verification means checking your answers with the original conditions or constraints set by the problem.

• For our drinks, we confirm \( x + y = 100 \) to see if we used all the pineapple juice, and indeed, \( 0 + 100 = 100 \).
• For the pomegranate concentrate, verify if \( 0.02 \times 0 + 0.04 \times 100 = 4 \). This also holds true.

Both of these checks confirm that the solution is correct and meets the problem's requirements, showing that all given resources are effectively utilized. Verifying solutions adds credibility to your results, ensuring no errors are made.