Problem 22 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ A chemist wishes to strengthen a mixture from \(10 \%\) alcohol to \(30 \%\) alcohol. How much pure elcohol should be added to 7 liters of the \(10 \%\) mixture?

Step-by-Step Solution

Answer

Add 2 liters of pure alcohol to achieve 30% concentration.

Step 1: Understand the Problem

We need to determine how much pure alcohol to add to a 7-liter mixture that is currently 10% alcohol, in order to increase the concentration to 30% alcohol.

Step 2: Set Up the Equation with Variables

Let \( x \) be the amount of pure alcohol in liters that needs to be added. The initial amount of alcohol is \( 0.1 \times 7 \) liters, because the solution is 10% alcohol. After adding \( x \) liters of pure alcohol, the total volume of the mixture becomes \( 7 + x \) liters.

Step 3: Write the Alcohol Content Equation

The total alcohol after adding \( x \) liters of pure alcohol will be \( 0.1 \times 7 + x \), and this should equal 30% of the new total mixture volume \( 7 + x \). Formulate the equation: \[ 0.1 \times 7 + x = 0.3 \times (7 + x) \]

Step 4: Simplify and Solve the Equation

Simplify the equation: \[ 0.7 + x = 2.1 + 0.3x \] Subtract \( 0.3x \) from both sides to get:\[ 0.7 + 0.7x = 2.1 \]Subtract 0.7 from both sides to obtain:\[ 0.7x = 1.4 \] Finally, solve for \( x \) by dividing both sides by 0.7:\[ x = 2 \]

Step 5: Verify the Solution Graphically

We can graph the equations \( y_1 = 0.1 \times 7 + x \) and \( y_2 = 0.3 \times (7 + x) \). The solution \( x = 2 \) should be the point where these two lines intersect, confirming that adding 2 liters of pure alcohol achieves the desired concentration.

Key Concepts

Mixture Problems

Mixture problems are a type of word problem that involves combining substances to achieve a desired composition. In this case, the task is to alter the concentration of an alcohol solution by adding more of one component to the existing mixture. When tackling these problems, the first step is understanding the goal, like increasing or decreasing concentration or achieving a specific mixture property.
To solve a mixture problem, identify:

• The original components and their concentrations.
• The amount or percentage of each in the initial mixture.
• What needs to be added or removed to achieve the desired mixture.

In our alcohol problem, we start with a 7-liter mixture containing 10% alcohol and need to increase this concentration to 30% using pure alcohol. Defining variables to represent unknowns, like the volume of pure alcohol to add, helps in structuring the solution.

Linear Equations

Linear equations are crucial for solving mixture problems as they provide a mathematical way to represent the relationships between different components of a mixture. In this exercise, we used a linear equation to calculate how much pure alcohol needs to be added.Setting up a linear equation involves:

• Defining a variable for the unknown quantity, such as the amount of pure alcohol.
• Expressing initial conditions and desired results mathematically.
• Equating the sum of initial quantities and added amounts to the final desired concentration.

In the exercise, let \( x \) be the liters of pure alcohol to add. The linear equation is formed as:\[ 0.1 \times 7 + x = 0.3 \times (7 + x) \]This represents the fact that the initial alcohol content plus pure alcohol added should be equal to 30% of the new total volume. Simplifying and solving the equation helps find the exact amount of pure alcohol needed, which is crucial for mixture problems.

Graphical Verification

Graphical verification offers a visual means to confirm the solution to a problem involving equations. This approach provides clarity and helps validate analytical results by illustrating how two conditions or equations meet visually on a graph.For this mixture problem:

• Plot the equation representing initial and added alcohol content.
• Plot the equation representing the desired final alcohol concentration.
• The intersection point of the graphs shows where the two conditions are equalized.

In our specific solution, we plotted the lines corresponding to:- \( y_1 = 0.1 \times 7 + x \) (alcohol content after adding \( x \) liters)- \( y_2 = 0.3 \times (7 + x) \) (desired concentration)Their intersection confirms that adding 2 liters of pure alcohol achieves the target 30% concentration. This graphical method corroborates the algebraic solution.