Problem 24 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ Acid Mixture How much water should be added to 20 liters of an \(18 \%\) acid solution to reduce the concentration to \(15 \%\) acid?

Step-by-Step Solution

Answer

Add 4 liters of water to reduce the concentration to 15% acid.

Step 1: Understand the Problem

We need to find the volume of water to add to 20 liters of an 18% acid solution to make a 15% solution.

Step 2: Set Up the Equation

Let the amount of water added be \( x \) liters. The amount of pure acid in the original solution is \( 0.18 \times 20 \) liters, which equals 3.6 liters. After adding water, the solution becomes \( 20 + x \) liters, and we want it to be 15% acid. So the equation is \( 0.15(20 + x) = 3.6 \).

Step 3: Solve the Equation

Expand the equation: \( 0.15(20 + x) = 3 \) which is \( 3 + 0.15x = 3.6 \). Solve for \( x \) by subtracting 3 from both sides: \( 0.15x = 0.6 \). Then, divide both sides by 0.15 to find \( x \): \( x = \frac{0.6}{0.15} = 4 \).

Step 4: Verify the Solution

Verify by substitution: Adding 4 liters of water makes the solution volume \( 20 + 4 = 24 \) liters. The acid concentration is \( \frac{3.6}{24} = 0.15 \), confirming the solution is now 15% acid.

Step 5: Support Graphically

Graphically, we can plot the concentration function \( C(x) = \frac{3.6}{20 + x} \). The graph should show \( C(x) = 0.15 \) when \( x = 4 \). This supports our analytical solution.

Key Concepts

Analytical Solutions

To tackle the problem of adjusting the concentration of an acid solution, analytical solutions provide a systematic approach. Through a series of logical steps, we break down a complex problem into solvable parts. In our given task, we needed to find out how much water to add to a 20-liter solution of 18% acid to dilute it to 15% acid. The analytical approach first involves understanding the quantities present, which requires calculating the amount of pure acid.
In this case, we calculated that the original solution has 3.6 liters of pure acid. Next, we built an algebraic equation based on this understanding: after adding water (let's call that quantity "x"), the total solution becomes \(20 + x\) liters. We aim for this mixture to be 15% acid, translating to the equation \(0.15(20 + x) = 3.6\).
Solving this straightforward algebraic equation helps us find the precise amount of water needed—4 liters in this instance. This method gives a clear, quantifiable result: a key feature of an analytical solution.

Graphical Verification

While analytical solutions provide an exact answer, graphical verification serves as a visual check. It helps confirm that the problem has been solved correctly. By plotting a graph of the relationship, we can see if our solution holds true visually. For this problem, the concentration of acid as water is added can be described by the function \(C(x) = \frac{3.6}{20 + x}\).
Graphing this function and identifying where it intersects with the desired concentration (15%) is crucial. We look for the point \(x = 4\) on the graph where the concentration meets 15%, affirming that the right amount of water was calculated. Such visual confirmation is a powerful tool in mathematics for students, bringing abstract equations to life and ensuring a deeper understanding.
This method not only validates our algebraic manipulation but also enhances comprehension by depicting how variables interact.

Algebraic Equations

Algebraic equations are the backbone of many mathematical solutions, translating real-world problems into mathematical language. In this scenario, the blending of solutions required setting up an algebraic equation to relate the original and desired concentrations of acid. We start by determining the existing amount of pure acid in the solution, given by multiplying the concentration by the volume: \(0.18 \times 20 = 3.6\) liters.
Next, as we add an unknown amount of water \(x\), the total volume changes, requiring us to set up the equation \(0.15(20 + x) = 3.6\). Solving such an equation involves expanding, rearranging terms, and solving for the unknown variable "x", which represents the liters of water needed.
Mastering such equations is crucial:

• They help solve a range of practical problems.
• They provide a quantifiable method for the solution.
• The result is precise, reliable, and testable.

Grasping algebraic equations is vital for tackling real-life challenges with mathematical strategies.