Problem 88 - Precalculus

Question

Orange juice producers use three varieties of oranges: Hamlin, Valencia, and navel. They want to make a juice mixture to sell at \(\$ 3.00\) per gallon. The price per gallon of each variety of juice is \(\$ 2.50, \$ 3.40,\) and \(\$ 2.80,\) respectively. To maintain their quality standards, they use the same amount of Valencia and navel oranges. Determine the quantity of each juice used to produce 1 gallon of mixture.

Step-by-Step Solution

Answer

Use \( \frac{1}{6} \) gallon Hamlin, \( \frac{5}{12} \) gallon Valencia, and \( \frac{5}{12} \) gallon Navel juice.

Step 1: Define Variables

Let's define the variables for the quantity of each orange juice variety used in the mixture. Let \( x \) be the amount (in gallons) of Hamlin juice, \( y \) be the amount of Valencia juice, and \( z \) be the amount of Navel juice. Given that the mixture is 1 gallon, we have:\[ x + y + z = 1 \]

Step 2: Set Up Cost Equation

The producers aim for the mixture to cost \( \$3.00 \) per gallon. Using the prices of each variety, we create the cost equation as follows:\[ 2.50x + 3.40y + 2.80z = 3.00 \]

Step 3: Use Quality Standard Equation

They use the same amount of Valencia and Navel oranges, which gives us:\[ y = z \]

Step 4: Substitute and Solve the System of Equations

Substitute the equation from Step 3 (\( y = z \)) into the equations from Steps 1 and 2. This gives us two equations:1. \( x + 2y = 1 \)2. \( 2.50x + 3.40y + 2.80y = 3.00 \)Simplify the second equation:\[ 2.50x + 6.20y = 3.00 \]Now, solve this system of equations. From (1):\[ x = 1 - 2y \]Substitute \( x = 1 - 2y \) into the second equation:\[ 2.50(1 - 2y) + 6.20y = 3.00 \]Simplify and solve for \( y \):\[ 2.50 - 5.00y + 6.20y = 3.00 \]\[ 1.20y = 0.50 \]\[ y = \frac{0.50}{1.20} = \frac{5}{12} \]

Step 5: Calculate Quantities of Hamlin, Valencia, and Navel Juices

With \( y = \frac{5}{12} \), and knowing \( y = z \), so \( z = \frac{5}{12} \). Substitute back into \( x = 1 - 2y \):\[ x = 1 - 2 \times \frac{5}{12} = 1 - \frac{10}{12} = \frac{1}{6} \]Therefore, the quantities used are:- Hamlin: \( x = \frac{1}{6} \) gallon- Valencia: \( y = \frac{5}{12} \) gallon- Navel: \( z = \frac{5}{12} \) gallon.

Key Concepts

Linear Equations

Linear equations are mathematical statements that express the equality of two expressions by using a straight line when you graph them. In the context of our exercise, linear equations help us understand the relationships between different quantities of orange juice in a mixture.
These equations are characterized by variables that are combined using addition, subtraction, and constant multiplication to form a linear relation. Here, we have two main linear equations: the total volume equation \[ x + y + z = 1 \] and the cost equation \[ 2.50x + 3.40y + 2.80z = 3.00. \]
Each variable (\( x, y, \) and \( z \)) represents a different type of orange juice, and their coefficients (like 2.50, 3.40, and 2.80) show how each variable contributes to the total.
When solved together, these linear equations describe the unique combination of the three types of orange juices needed to meet specific constraints, such as volume and cost.

Substitution Method

The substitution method is a strategy used to solve systems of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equations. This reduces the number of equations and variables, simplifying the system.
In our problem, we start by recognizing that the amount of Valencia and Navel juice must be equal, represented by the equation \( y = z \). By substituting \( z \) with \( y \) in the equations, we simplify our system:

• \( x + 2y = 1 \)
• \( 2.50x + 6.20y = 3.00 \).

Then, we substitute \( x = 1 - 2y \) from the first simplified equation into the second, which allows solving directly for \( y \).
This step-by-step narrowing down is efficient and solves the system of equations by reducing it to one variable.

Cost Equations

Cost equations are crucial when dealing with problems involving budgeting and costing, like the production of juice. These equations allow us to calculate the total cost based on the cost and amount of each ingredient.
In this exercise, the cost equation \[ 2.50x + 3.40y + 2.80z = 3.00 \] reflects the goal of achieving an overall price of \( \$3.00 \) per gallon of juice mix. The coefficients 2.50, 3.40, and 2.80 represent the cost per gallon of Hamlin, Valencia, and Navel juices, respectively.
By setting up this equation, we can see how much each type of juice costs and how their proportions influence the final cost of the mixture. Solving the cost equation in conjunction with other constraints (like the volume and quality standards) ensures that the solution is both practical and satisfies all the problem's requirements.

Variables in Equations

In mathematics, variables act as placeholders that can hold different values and are crucial in forming equations.
In our exercise, we define three variables: \( x, y, \) and \( z \). Each represents the gallons of Hamlin, Valencia, and Navel juices used.
Variables allow us to explore different possibilities and conditions, such as maintaining equal quantities of Valencia and Navel juices (\( y = z \)) or ensuring the total volume is exactly one gallon (\( x + y + z = 1 \)).
By strategically manipulating these variables within equations, we can reveal the precise amounts of each type of juice needed to create the desired mixture. Understanding the role and flexibility of variables makes it easier to find optimized solutions and apply similar techniques to a variety of problems.