Problem 124 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The function$$ f(x)=\frac{1.96 x+3.14}{3.04 x+21.79} $$ models the fraction of nonviolent prisoners in New York State prisons x years after 1980 . I can conclude from this equation that over time the percentage of nonviolent prisoners will exceed 60 \%.

Step-by-Step Solution

Answer

Without specific numerical solutions provided, we cannot definitively say if the statement makes sense without doing the calculations. However, if the solution to the equation in step 3 is a positive real number, then the statement makes sense. Otherwise, it does not make sense.

Step 1: Understand the Statement

The function \( f(x)=\frac{1.96 x+3.14}{3.04 x+21.79} \) is given as the model describing the fraction of nonviolent prisoners in New York State prisons \( x \) years after 1980.

Step 2: Testing the Statement

To determine whether the fraction will exceed 60%, convert 60% to decimal form which gives 0.6. Now set the function \( f(x) \) equal to 0.6 and solve for \( x \).

Step 3: Solve for x

Set up the equation \( \frac{1.96 x+3.14}{3.04 x+21.79} = 0.6 \) and solve for \( x \).

Step 4: Analyze the Result

If the solution for \( x \) is a real number, it implies that at some year after 1980, the percentage of nonviolent prisoners does exceed 60%. If the solution is not a real number, it implies the percentage will not exceed 60% within the given context.

Key Concepts

Fraction

Fractions are an essential part of mathematics, especially when dealing with rational functions. A fraction represents a part of a whole, consisting of two numbers: the numerator (top number) and the denominator (bottom number). In the context of the function \( f(x)=\frac{1.96x+3.14}{3.04x+21.79} \), this entire expression can be considered a complex fraction that models a real-world scenario.

Here's how it breaks down:

• The numerator, \(1.96x + 3.14\), represents the part of nonviolent prisoners over time.
• The denominator, \(3.04x + 21.79\), states the total number, capturing both nonviolent and possibly violent prisoners.

This expression models how the fraction of nonviolent prisoners changes as time progresses from the year 1980. It gives a snapshot of how the composition of the prison population changes over time after 1980.

Understanding fractions in such models helps in predicting and analyzing trends within given contexts. Just as we analyze a simple fraction like \( \frac{1}{2} \), we analyze rational functions to understand dynamic systems.

Percentage

Percentages offer an approachable way to understand parts of a whole as they allow for easy comparison and interpretation. In our problem, the goal is to determine if the percentage of nonviolent prisoners in the model could ever exceed 60%.

To analyze this, the procedural step is to convert the percentage into a decimal form. So, 60% transforms into 0.6 for ease of calculation. Then, we set the fraction \( f(x)=\frac{1.96x+3.14}{3.04x+21.79} \) equal to this decimal to solve for \( x \). The equation becomes:

• \( \frac{1.96x+3.14}{3.04x+21.79} = 0.6 \)

This setup helps determine if a realistic solution for \( x \) exists where the function equals 0.6, meaning 60% of the prison population could be nonviolent. Solving the equation will tell if such a scenario is possible in the context of the years following 1980.

Using percentages makes interpreting data intuitive and effective, as people often understand percentages better than fractions.

Modeling

Modeling with rational functions like the one in our exercise is a powerful way to represent real-world behaviors mathematically. It involves using functions to depict relationships between variables over a continuum of values. Here, the function \( f(x)=\frac{1.96x+3.14}{3.04x+21.79} \) models changes in the percentage of nonviolent prisoners over time.

Why do we use such models?

• They help predict future behaviors and outcomes based on current data.
• They make it possible to simulate scenarios for decision-making purposes.
• They allow us to understand complex systems by breaking them down into simpler relationships.

In this context, the model tests a hypothesis: whether nonviolent prisoners would constitute a significant majority in the future. To validate this, solving the equation helps find if any point in time after 1980 meets or exceeds the 60% threshold for nonviolent prisoners.

Thus, mathematical modeling becomes a fundamental tool for planners, policymakers, and researchers to make data-driven decisions.