Problem 78 - College Algebra and Calculus: An Applied Approach

Question

Use the following information. From 1998 to 2005, the annual credit \(y\) (in billions of dollars) extended to consumers in the United States (other than real estate loans) can be approximated by the equation \(y=129.51 t+320.5, \quad 8 \leq t \leq 15\) where \(t\) is the year, with \(t=8\) corresponding to 1998. Use the model to predict the year in which the credit extended to consumers will be about \(\$ 2.9\) trillion.

Step-by-Step Solution

Answer

After performing the calculations, find that \(t \approx 16\). Adding this to 1990 gives the year 2006, so the model predicts that the credit extended to consumers will be about \$2.9 trillion around the year 2006.

Step 1: Convert Trillions to Billions

To keep the units consistent, convert the credit in trillions to billions. 1 trillion is 1000000 billion, so \(2.9\) trillion is \(2.9 * 1000000 = 2900000\) billion dollars.

Step 2: Setting Up the Equation

The problem involves predicting the year when the annual credit extended reaches \(\$ 2.9\) trillion, or \(2900000\) billion. Plug this value into the equation, so it becomes \(2900000 = 129.51t + 320.5\).

Step 3: Solving for t

Re-arrange the equation to solve for \(t\), which involves subtracting \(320.5\) from each side, then dividing by \(129.51\). The new equation becomes \((2900000 - 320.5) / 129.51 = t\).

Step 4: Calculate t

Perform the operation to find the value of \(t\). Remember, \(t=8\) corresponds to 1998, so to find the actual year, add the result to 1990.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. One variable, known as the dependent variable, is considered to be predicted by one or more independent variables. The simplest form of the linear regression equation is given as \( y = mx + b \), where \( y \) is the predicted value, \( m \) is the slope of the line, \( x \) is the independent variable, and \( b \) is the y-intercept. This equation allows for making predictions about the dependent variable based on new values of the independent variable.

In the case of the credit prediction model presented in the exercise, the annual credit extended to consumers (\(\ y \)) is the dependent variable predicted by the independent variable \( t \), representing the year. The coefficients \(129.51\) being the slope, and \(320.5\) the intercept, were likely determined through a linear regression analysis using historical data from 1998 to 2005.

Improving accuracy in a linear regression model could involve examining and including more predictors, ensuring the linearity of data, checking for outliers, and considering the temporal aspects such as inflation or economic cycles which might impact consumer credit trends.

Credit Market Trends

Credit market trends reflect the patterns and changes in borrowing and lending behaviors over time. These trends are influenced by various factors, including economic conditions, interest rates, inflation, and consumer confidence. It is crucial to understand these trends to predict future credit market movements effectively.

Anticipating changes in the credit market can help financial institutions manage risk and create strategies to meet consumer demand. In the exercise provided, the annual credit extended to consumers follows a linear trend that signifies a steady increase over time from 1998 to 2005. However, to enhance the model, one should examine other periods and additional factors that could cause fluctuations in consumer borrowing behaviors. For instance, incorporating data related to economic downturns or booms would enrich the model's predictive capability and provide a more realistic projection.

Algebraic Modeling

Algebraic modeling involves creating mathematical models that translate real-world scenarios into algebraic expressions or equations. These models can analyze and make predictions about different phenomena. In algebraic modeling, variables represent quantities we want to investigate, and equations connect these variables in a way that mirrors the real-world relationships.

In the context of the original exercise, the equation \(y = 129.51 t + 320.5\) is an algebraic model that represents the relationship between time and the annual credit extended to consumers. This model simplifies a complex reality into a form that can be easily worked with and understood. When improving an algebraic model, it's essential to verify that the model accurately represents the data and includes relevant variables. Also, remember to keep the model simple enough to be functional, avoiding unnecessary complexity that could make the model less interpretable and reduce its predictive power.

Understanding the underlying principles of algebraic modeling helps students to make meaning of the steps provided in the credit prediction problem and apply them to various other predictive scenarios in finance and economics.