Problem 116 - College Algebra Essentials

Question

The bar graph indicates that the percentage of fi rst-year college students expressing antifeminist views declined after 1970. CAN'T COPY THE GRAPH The function $$f(x)=-4.82 \ln x+32.5$$ models the percentage of first-year college women, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969 a. Use the function to find the percentage of first-year college women expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college women who will express antifeminist views in \(2015 .\) Round to one decimal place.

Step-by-Step Solution

Answer

The percentage of women expressing anti-feminist views in 2008 is determined by the function \(f(x)= -4.82 \ln x + 32.5\) for \(x=39\). Similarly, for the year 2015, the value of \(x\) is 46. Without the graph, it cannot be verified whether these function values overestimate or underestimate the true values.

Step 1: Calculate the Value for 2008

To calculate the percentage of women expressing anti-feminist views in 2008, the value of \(x\) has to be determined in relation to 1969. Therefore, for 2008, \(x \) would be 2008 - 1969 = 39. Plug this value into the function: \(f(39)= -4.82 \ln (39) + 32.5 \)

Step 2: Calculate the Value for 2015

Similar to step 1, for 2015, \(x \) would be 2015 - 1969 = 46. Plug this into the function to get: \(f(46)= -4.82 \ln(46) + 32.5 \)

Step 3: Analysis of the results

After finding the results for each year, compare these results to the data (if provided) on the graph to determine if the function overestimates or underestimates the values. Without the graph, these interpretations cannot be determined.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental part of algebra and calculus, often used to describe phenomena that decrease exponentially or exhibit rapid changes initially that slow down over time. The function provided in the exercise is a logarithmic function: \( f(x) = -4.82 \ln x + 32.5 \). This type of function is essential for modeling various real-world behaviors where a quantity decreases at a rate proportional to its current value, such as radioactive decay or the cooling of a hot object.

In our specific exercise, the logarithmic function models a social trend about college students' attitudes. Here, \( \ln x \) represents the natural logarithm of \( x \), which reveals how the percentage decreases as the years pass. The coefficient \(-4.82\) indicates the rate at which the percentage of students expressing antifeminist views declines. A negative coefficient suggests a decreasing trend over time.

Understanding this function requires recognizing how changes in \( x \) (years since 1969) affect \( f(x) \). A crucial aspect of logarithmic functions is their slopes changing more gradually as \( x \) increases, which makes them useful for long-term projections like predicting social attitudes.

Mathematical Modeling

Mathematical modeling is a process that involves creating a mathematical representation of a real-world scenario, which allows for predictions and analyses. In this exercise, the model \( f(x) = -4.82 \ln x + 32.5 \) is designed to simulate the percentage of first-year college women with antifeminist views over time.

Modeling involves several steps:

• Defining the Problem: The issue here is to understand and predict changes in antifeminist views among college students.
• Formulating the Model: The function is constructed based on historical data, using logarithms to account for the gradual decline.
• Making Predictions: By plugging values of \( x \) corresponding to specific years, predictions are made for 2008 and 2015.
• Interpreting the Results: Comparison with any provided data (like a graph) can validate the model's accuracy.

Such models are invaluable for data analysis, enabling researchers to understand trends and prepare for future scenarios. Although the given function might not account for all factors influencing students' attitudes, it provides a basis for estimating trends.

Data Analysis

Data analysis involves examining, cleaning, and interpreting data to extract meaningful insights and make informed decisions. In the context of this exercise, data analysis is used to compare calculated values of antifeminist views with actual historical data to determine accuracy.

Several steps facilitate effective data analysis:

• Data Collection: Original observations or collected data, such as survey results, serve as the foundation for analysis.
• Calculation: Using the function to find percentages for specific years, like 2008 and 2015, helps in drawing comparisons.
• Comparison: These calculated values are compared to data points (like those on a graph) to establish if the model overestimates or underestimates real values.
• Interpretation: Understanding the degree of variance between model predictions and actual data assists in refining the model or acknowledging its limitations.

Data analysis provides a framework for assessing the function's reliability and usefulness. By understanding discrepancies, analysts can highlight areas where the model could improve, thus sharpening predictive capabilities for future studies.