Problem 112 - College Algebra with Modeling and Visualization

Question

Mixing Antifreeze \(\quad\) A radiator holds 5 gallons of fluid. If it is full with a \(15 \%\) solution, how much fluid should be drained and replaced with a \(65 \%\) antifreeze mixture to result in a \(40 \%\) antifreeze mixture?

Step-by-Step Solution

Answer

Drain and replace 2.5 gallons with 65% antifreeze solution.

Step 1: Understanding the Problem

A radiator contains a 15% antifreeze solution and is full at 5 gallons. We need to determine the amount of this solution that should be drained and replaced with a 65% antifreeze solution to achieve a 40% antifreeze mixture.

Step 2: Setting Up Equations

Let's call the amount of fluid to be drained and replaced 'x' gallons. Initially, there are 5 gallons with 15% antifreeze, so 0.15 * 5 gallons of antifreeze is present. The solution after exchanging x gallons will have (5-x) gallons of the 15% solution and x gallons of the 65% solution.

Step 3: Expressing Components in Terms of x

After draining x gallons, the remaining fluid has a total antifreeze content of:- 15% of (5-x) gallons: \[0.15 \times (5-x)\]- 65% of x gallons: \[0.65 \times x\]The total volume remains 5 gallons, and the total antifreeze amount should ensure a 40% concentration of the total 5 gallons, or 0.40 * 5.

Step 4: Forming the Equation

The equation to represent the total antifreeze in the final solution is:\[0.15(5-x) + 0.65x = 0.40 \times 5\]This equation represents the concentration balance before and after the change.

Step 5: Solving the Equation

First, calculate the total antifreeze needed:\[0.40 \times 5 = 2 \, \text{(gallons of pure antifreeze)}\]Set up the equation:\[0.15(5-x) + 0.65x = 2\]Simplify and solve for x:\[0.75 - 0.15x + 0.65x = 2\]Combine terms:\[0.75 + 0.50x = 2\]Subtract 0.75 from both sides:\[0.50x = 1.25\]Divide by 0.50:\[x = 2.5\]

Step 6: Conclusion

Therefore, 2.5 gallons need to be drained and replaced with a 65% antifreeze solution to achieve a 40% solution in the radiator.

Key Concepts

Mixture Problems

Mixture problems are a common type of algebra word problem. They involve combining different substances in varying proportions to achieve a desired consistency or concentration. In the context of our exercise, the radiator mixture involved two antifreeze solutions with different concentrations.

Usually, the objective is to either find the amount of a component to add or remove to reach a target concentration. This type of problem requires understanding how parts of a mixture contribute to the whole, and typically involves making calculations based on percentages. By carefully setting up an equation that balances the quantities and concentrations of each part, you can determine the necessary amount of each component.

For instance, in our antifreeze problem, we are tasked with adjusting the existing mixture to reach a certain concentration. By keeping the total volume constant and focusing on the concentration percentages, we can calculate how much of each type of solution we need to reach the desired mixture.

Linear Equations

Linear equations are mathematical expressions that model the relationships between variables in a linear form. They are building blocks for solving problems like mixture problems, where the solution must balance different components proportionally.

In a mixture problem, each type of solution can be represented as a linear equation. This is particularly helpful for determining how much of one component needs to be added or removed to achieve a desired result. The beauty of linear equations is in their simplicity; they form straight lines when graphed, which makes them easy to solve and predict.

The key step is setting up the equation, which typically involves identifying the total concentration needed and equating it to the sum of individual components. In the radiator example, we used a simplification of a linear equation to solve for the variable representing how much to drain and replace. This equation then allows for straightforward manipulation to uncover the desired quantity of the high-concentration solution.

Percent Concentration Problems

Percent concentration problems are crucial for understanding how to accurately mix solutions with specific properties. These problems involve the use of percentages to determine how much of each component is present in a mixture. This is particularly common in chemistry and everyday situations like cooking and fuel mixing.

The exercise's challenge was to manage the concentration of antifreeze in a radiator. To achieve the desired concentration, you need to calculate how the amounts and concentrations of the components add up to the final mixture. The percent concentration is crucial for determining exactly how much of each solution is required.

To solve the antifreeze concentration problem, we calculated the contribution of each part to the total concentration. By understanding the percentages, you can apply a straightforward mathematical operation like multiplying by the overall total to get the desired concentrations. Solving percent concentration problems involves carefully balancing components to meet a specific goal, much like aligning puzzle pieces so that they fit perfectly together.