Problem 23 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ Saline Solution Mixture How much water should be added to 8 milliliters of \(6 \%\) saline solution to reduce the concentration to \(4 \%\) saline?

Step-by-Step Solution

Answer

Add 4 milliliters of water to reduce the concentration to 4%.

Step 1: Define the Variables

Let \( x \) be the amount of water to be added in milliliters. We start with 8 milliliters of a 6% saline solution. Our goal is to find \( x \) so that the new solution has a concentration of 4% saline.

Step 2: Set up the Equation

The initial amount of salt in the solution is \( 0.06 \times 8 = 0.48 \) milliliters. After adding \( x \) milliliters of water, the total volume of the solution becomes \( 8 + x \) milliliters. We want the salt concentration to be 4%, so the equation becomes: \( \frac{0.48}{8+x} = 0.04 \).

Step 3: Solve the Equation

To solve \( \frac{0.48}{8+x} = 0.04 \), first multiply both sides by \( 8 + x \) to get \( 0.48 = 0.04(8 + x) \). This simplifies to \( 0.48 = 0.32 + 0.04x \). Subtract 0.32 from both sides to get \( 0.16 = 0.04x \). Dividing both sides by 0.04 gives us \( x = 4 \).

Step 4: Verify the Solution Graphically

Graph the initial concentration \( C = \frac{0.48}{8+x} \) and see where it intersects with \( y = 0.04 \). Confirm the intersection occurs at \( x = 4 \), which verifies through a graphical approach that \( x = 4 \) is indeed correct.

Key Concepts

Analytical Problem Solving

Analytical problem solving involves breaking down the problem into smaller, manageable parts and using mathematical tools to work through them. In the case of the saline solution mixture problem, you're asked to change the concentration of a salt solution by adding water. This approach revolves around understanding the proportionality of solution concentration and the effect of dilution.

Begin by identifying key elements like the initial condition (8 mL of a 6% solution), and what needs to be achieved (4% solution after adding water). This requires defining a variable, in this case, let’s call it \(x\) for the water to be added. We know that adding pure water doesn't change the amount of salt but increases the total volume. Hence, the analysis is grounded in setting up an equation based on these initial and required conditions. This systematic method helps to focus on the tangible elements of the problem and move towards the solution logically.

Graphical Verification

Graphical verification is a powerful tool to support your analytical findings. It allows you to visually interpret the relationship between variables. In our problem, we need to confirm that adding a certain amount of water reduces the saline concentration as expected.

Start by plotting the function that represents the concentration: \( C = \frac{0.48}{8 + x} \). Next, draw a horizontal line at \( y = 0.04 \), representing the desired concentration of 4%. The point where the concentration curve intersects the horizontal line indicates the necessary amount of water. If your solution of \( x = 4 \) milliliters is correct, the graph will visually confirm it at the intersection point (4, 0.04). Graphical tools can provide reassurance for complex calculations by offering a different perspective on the problem solution.

Equation Setup

Setting up the right equation is a cornerstone of solving mathematical problems effectively. Here, we start by determining the actual quantity of salt in the original solution. Since 6% of 8 milliliters is used, calculate this by \( 0.06 \times 8 = 0.48 \) milliliters of salt. Next, you need to represent the amount of this salt in the new total volume after dilution.

The equation to express this relation is \( \frac{0.48}{8 + x} = 0.04 \), which implies that even after adding water, the salt per total volume maintains a specific concentration. Simplifying this equation involves multiplying through to clear the fraction, followed by isolating \( x \). Each step in setting up this equation logically corresponds to aspects of concentration and volume, leading naturally to the solution \( x = 4 \). Mastering equation setup is crucial for efficiently dealing with dilution problems and ensures that the basic principles of proportions and balance are preserved through calculations.