Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. Antifreeze Mixture An automobile radiator contains a 10 -quart mixture of water and antifreeze that is \(40 \%\) antifreeze. How much should the owner drain from the radiator and replace with pure antifreeze so that the liquid in the radiator will be \(80 \%\) antifreeze?

Step-by-Step Solution

Answer

Drain approximately 6.67 quarts and replace with pure antifreeze.

Step 1: Understand the Problem

We have a 10-quart mixture that is 40% antifreeze. Our goal is to adjust this mixture so that it becomes 80% antifreeze by draining out some of the existing mixture and adding pure antifreeze in its place.

Step 2: Set up Variables and Equations

Let \( x \) represent the amount of the current mixture to be drained and replaced with pure antifreeze. After replacing \( x \) quarts, the radiator will still have 10 quarts total, now consisting of more antifreeze. The remaining mixture will contain \( 0.4(10-x) \) quarts of antifreeze.

Step 3: Establish the Antifreeze Equation

After draining \( x \) quarts and adding \( x \) quarts of pure antifreeze, the new amount of antifreeze in the radiator will be \( 0.4(10-x) + x \). We want this to be 80% of the total 10 quarts.

Step 4: Set up the Final Equation

The equation representing 80% antifreeze in the 10-quart mixture is: \[ 0.4(10-x) + x = 0.8 imes 10 \]

Step 5: Solve the Equation

Start by expanding the equation: \[ 4 - 0.4x + x = 8 \] Combine like terms: \[ 4 + 0.6x = 8 \] Subtract 4 from both sides: \[ 0.6x = 4 \] Divide by 0.6: \[ x = \frac{4}{0.6} = \frac{40}{6} = \frac{20}{3} \approx 6.67 \]

Step 6: Verify the Solution Graphically

Consider the line equation from the mixture problem plotted on a graph where the x-axis represents quarts drained and y represents the percentage of antifreeze. The intersection point with the horizontal line y=80% should be approximately \(x = 6.67\), confirming the solution.

Key Concepts

Percentages in Mixtures

The concept of percentages in mixtures is essential when dealing with problems like the antifreeze mixture described. Here, we have a certain composition of antifreeze in the radiator mixture, given as a percentage, and we want to adjust it. Imagine you have a simple cocktail of water with a specific amount of antifreeze. This percentage tells us how much of the solution is made up of antifreeze.
To solve such problems, follow these steps:

• Identify the initial composition percentage (in this case, 40% antifreeze).
• Determine the desired composition percentage (targeting 80% antifreeze).
• Recognize the total volume of the mixture to maintain, here at 10 quarts.

The task then becomes determining how much of the original mixture to drain and replace with pure antifreeze to achieve the targeted concentration. This requires a good understanding of percentages and how they apply to mixtures.

Linear Equations

Linear equations form the backbone of solving algebraic mixture problems. In the antifreeze exercise, we establish a linear equation to calculate the necessary change. Here's how it works:

• Let \( x \) be the amount of mixture to drain, assumed to be the same added back as pure antifreeze.
• The remaining antifreeze after draining \( x \) quarts is \( 0.4(10-x) \) quarts.
• The equation is set up to reflect the final condition of 80% antifreeze: \[ 0.4(10-x) + x = 8 \] which simplifies to a linear form.

Upon rearranging and solving these terms, you find \( x = \frac{20}{3} \) or approximately 6.67 quarts. Linear equations are pivotal for determining relationships and necessary changes in simple mixture problems.

Graphical Solution Verification

Graphical solution verification serves as a visual confirmation of the analytical solution derived from linear equations. When we graph the relationship, it provides a clear intersection point that should align with the calculated value.

• Plot a graph where the x-axis represents quarts drained and the y-axis the percentage of antifreeze.
• The line equation \( 0.4(10-x) + x \) forms a linear graph of antifreeze percentage against quarts drained.
• Check where this graph intersects with the horizontal line at y=80%.

The x-value at this intersection point should match the solution from your equation, providing a solid check. This visual method reassures the reliability of your calculation by showing the actual working range and relationships, making it easier to understand and trust.