Problem 51 - A Graphical Approach to Precalculus with Limits

Question

A sparkling-water distributor wants to make up 300 gallons of sparkling water to sell for \(\$ 6.00\) per gallon. She wishes to mix three grades of water selling for \(\$ 9.00, \$ 3.00,\) and \(\$ 4.50\) per gallon, respectively. She must use twice as much of the \(\$ 4.50\) water as the \(\$ 3.00\) water. How many gallons of each should she use?

Step-by-Step Solution

Answer

120 gallons of \(\$9.00\) water, 60 gallons of \(\$3.00\) water, 120 gallons of \(\$4.50\) water.

Step 1: Define Variables

Let's define the variables for the quantities of each type of water. Let \( x \) be the number of gallons of \( \\(9.00 \) per gallon water, \( y \) be the number of gallons of \( \\)3.00 \) per gallon water, and \( z \) be the number of gallons of \( \$4.50 \) per gallon water.

Step 2: Set Up Equations Based on Total Volume

The distributor wants to mix 300 gallons of water. Therefore, we can set up the equation: \[ x + y + z = 300 \]

Step 3: Set Up Equations Based on Total Value

The total value of the mixed water needs to be \( 300 \times \\(6.00 = \\)1800 \). Therefore, the equation for the total value is: \[ 9x + 3y + 4.5z = 1800 \]

Step 4: Incorporate The Given Ratio Condition

According to the problem, the amount of \( \\(4.50 \) water \( z \) is twice the amount of the \( \\)3.00 \) water \( y \). We express this as: \[ z = 2y \]

Step 5: Substitute and Solve

Substitute \( z = 2y \) in the equations:1. Substitute in the volume equation: \[ x + y + 2y = 300 \] Simplifying gives: \[ x + 3y = 300 \]2. Substitute in the value equation: \[ 9x + 3y + 4.5(2y) = 1800 \] Simplifying gives: \[ 9x + 3y + 9y = 1800 \] \[ 9x + 12y = 1800 \]

Step 6: Solve Simultaneous Equations

Now, solve the system of equations:- From \( x + 3y = 300 \), we have \( x = 300 - 3y \).- Substitute \( x = 300 - 3y \) into \( 9x + 12y = 1800 \): \[ 9(300 - 3y) + 12y = 1800 \] \[ 2700 - 27y + 12y = 1800 \] \[ 2700 - 15y = 1800 \] \[ 900 = 15y \] \[ y = 60 \]Find \( x \) using \( x = 300 - 3y \):- \( x = 300 - 3(60) = 300 - 180 = 120 \)Find \( z \) using \( z = 2y \):- \( z = 2(60) = 120 \).

Step 7: Verify Solution

Verify for correctness:- Total volume: \( 120 + 60 + 120 = 300 \)- Total value: \( 9(120) + 3(60) + 4.5(120) = 1080 + 180 + 540 = 1800 \)The solution satisfies both the volume and value equations.

Key Concepts

Linear Equations

Linear equations are equations that create a straight line when graphed. They are made with one or more variables where each term is either a constant or the product of a constant and a single variable.
In the problem of the sparkling-water distributor, linear equations help us figure out the mixture of different grades of water. For example, the equation \(x + y + z = 300\) is a linear equation that represents the total amount of water (300 gallons) she wants to make.
Linear equations in this context are useful because they allow us to model the situation with constraints, such as the total volume and total cost of the water mixture.

Substitution Method

The substitution method involves solving one of the linear equations for one variable and then substituting that expression into another equation. This technique helps reduce the number of variables, making the system of equations easier to solve.
In our example, we substitute \(z = 2y\) from the condition given in the problem into the volume equation \(x + y + z = 300\). Doing so reduces the equation to \(x + 3y = 300\), which has only two variables instead of three.
This simplifies the process of solving the equations, enabling us to find values for \(x\), \(y\), and \(z\) that satisfy both the original conditions.

Value Equation

The value equation represents the total monetary value of the water mixture. In this case, each type of water has a different price per gallon. Therefore, the value equation ensures that the overall cost of the mixtures fits the target of creating water worth \(\$6.00\) per gallon of the 300 gallons.
The equation, \(9x + 3y + 4.5z = 1800\), calculates the value by multiplying the quantity of each type of water by its cost per gallon and adding these together.
It helps confirm that the selected quantities satisfy the economic expectations of the distributor, ensuring the price condition is intertwined with the volume condition.

Volume Equation

A volume equation ensures that the total amount of material satisfies the required output. In this case, the distributor wants to mix 300 gallons of water, which is represented by the equation \(x + y + z = 300\).
This equation is crucial because it keeps track of the total gallons of different water grades, ensuring that the solution aligns with the physical requirement of the total mixture.

• The equation establishes the constraint that all types combined must equal 300 gallons.
• It serves as a base for adjusting the quantities of the different water grades within this constraint through the substitution method and other linear equations.

Ultimately, this balance is vital for achieving the desired product mix.