Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Work Exercises \(1-6\) without pencil and paper. Do not use a calculator. Suppose two acid solutions are mixed. One is \(26 \%\) acid and the other is \(32 \%\) acid. Which one of the following concentrations cannot possibly be the concentration of the mixture? C. \(30 \%\) B. \(28 \%\) D. \(31 \%\) A. \(36 \%\)

Step-by-Step Solution

Answer

Concentration 36% cannot be the mixture's concentration.

Step 1: Understanding the Problem

We have two solutions with known acid concentrations: one at 26% and the other at 32%. We need to determine which listed concentration cannot be the result when these two solutions are mixed together.

Step 2: Determine Possible Range

When two solutions are mixed, the concentration of the resulting mixture should fall between the two given concentrations, assuming we mix any amount of each solution. Thus, the possible concentration range for the mixture is between 26% and 32%.

Step 3: Analyze Each Option

Examine the mixture possibilities: - Option A: 36% is outside the range 26%-32%, so it cannot be a result of mixing these two solutions. - Options B (28%), C (30%), and D (31%) are all within the possible range.

Key Concepts

Acid Concentration

Acid concentration refers to the percentage of acid present in a solution compared to the whole volume. In this exercise, we are looking at two acid solutions: one with a 26% acid concentration and the other with a 32% acid concentration. Understanding how to work with percentages is crucial.
The percentage indicates the proportion of acid in a 100-part solution.
So, a 26% solution means 26 parts acid and 74 parts another substance, typically water, making up a total of 100 parts.
Likewise, in a 32% solution, 32 parts are acid.

• Why It's Important: Knowing the acid concentration helps determine the strength and behavior of the acid in chemical applications.
• Scenario Usage: When acids need to be diluted or concentrated for industrial or laboratory processes, understanding concentrations is essential.

Solution Mixing

Solution mixing involves combining two or more solutions to create a new solution. When mixing acidic solutions, the resulting concentration is influenced by the concentrations and volumes of the solutions mixed.

Imagine pouring two different liquids into one container. Once mixed thoroughly, the new solution will have properties determined by the original volumes and concentrations.

• Volume Influence: If you mix equal volumes of the 26% and 32% solutions, the concentration of the new mixture will average between those two percentages.
• Concentration Influence: The exact concentration of the new mixture depends on the proportion of each solution used.

The key is finding a balance in quantities that predictably affect the final mixture's outcome.

Percentage Range

The percentage range is the interval between the smallest and largest possible concentrations when mixing solutions. In our exercise, the highest and lowest concentrations are 32% and 26% respectively.
This indicates that any resulting mixture should settle within this range as the concentrations cannot exceed or fall below either starting value.

• Real-world Application: This concept is used in quality control processes to ensure that mixtures meet specifications before use or sale.
• Critical Thinking: By identifying the range, one can quickly deduce which options are or are not feasible in mixture problems.

In our example, a 28%, 30%, or 31% concentration fits within our range, but a 36% concentration does not, making it impossible.

Precalculus

Precalculus lays the groundwork for understanding complex mathematical concepts and relationships. It often deals with functions and their behaviors, just like how mixture problems use functions to determine outcomes.
In problems like these, precalculus tools help you form and solve equations that describe the mixing process.

• Core Skills: Understanding and manipulating percentages and rates are fundamental skills honed in precalculus.
• Mathematical Modeling: Developing scenarios and conditions under which certain outcomes occur by employing equations and inequalities.

This background helps you recognize how percentages interact within a given range and how each percentage move impacts the solution's concentration.