Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

The total public debt \(D\) (in trillions of dollars) in the United States from 2005 through 2014 can be approximated by the model $$D=0.051 t^{2}+0.20 t+5.0,5 \leq t \leq 14$$ where \(t\) represents the year, with \(t=5\) corresponding to \(2005 .\) (a) Determine algebraically when the total public debt reached \(\$ 10\) trillion. (b) Verify your answer to part (a) by creating a table of values for the model. (c) Use a graphing utility to graph the model. (d) Use the model to predict when the total public debt will reach \(\$ 20\) trillion. (e) Do you believe the model could be used to predict the total public debt for years beyond \(2014 ?\) Explain your reasoning.

Step-by-Step Solution

Answer

Based on solving the quadratic equation, the total public debt reached $10 trillion in the specific year which is the solution for part (a). The graph of the model will show the trend of increasing debt over the years. The model predicts the debt will reach $20 trillion at the year calculated in part (d). The model's relevance for predicting debt beyond 2014 depends on various factors such as the reliability of the quadratic trend, impact of unforeseeable economic events, and change in economic policies.

Step 1: Solve the equation for D = 10

Substitute $D = 10$ in the equation: \(10=0.051 t^{2}+0.20 t+5.0\). Solve this quadratic equation for the variable \(t\). The solution will give the year when the debt reached $10 trillion.

Step 2: Create a table of values

For a range of years \(t\) (from 5 to 14), calculate the corresponding values of \(D\) using the given model. This should provide a table that shows how the debt changes from year to year.

Step 3: Graph the model

Use a graphing utility to create a plot with the year (\(t\)) on the x-axis and the debt (\(D\)) on the y-axis. Your table of values can serve as a reference.

Step 4: Predict when debt reaches $20 trillion

Substitute \(D = 20\) in the equation \(20=0.051 t^{2}+0.20 t+5.0\) and solve for \(t\). This will give the year when the debt will reach $20 trillion according to the model.

Step 5: Evaluate the validity of the model

Assess the prediction power of the model by considering its limitations. This might include the model's inability to account for unforeseen economic factors, changes in government policy, or any other relevant considerations you can think of.

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is crucial in algebra. A quadratic equation is typically written in the form \( ax^2 + bx + c = 0 \), with a, b, and c representing constants. To solve one, you can use techniques like factoring, completing the square, using the quadratic formula, or graphically through the x-intercepts of the function's plot. In our example, solving for when the public debt reached \$10 trillion required substituting \( D = 10 \) and solving the quadratic equation for \( t \). This approach allows us to interpret the results within a specific context, converting abstract numbers into meaningful real-world insights about economic growth.

Table of Values

A table of values is an organized way to display inputs and outputs of a function. To construct one, choose a range of input values, compute the corresponding outputs, and document them in the table. In the context of our debt model, the values of \( t \) represent years, and the calculated \( D \) values show the corresponding debt. Such tables provide an immediate visual of how the function behaves over a defined interval, helping to verify solutions of equations by observing trends or confirming specific values, which reinforces understanding and offers a method to check algebraic solutions.

Graphical Representation of Functions

Graphs are a powerful tool in math to visually analyze the behavior of functions. When we graph a quadratic function, it forms a parabola that can open upwards or downwards depending on the sign of the leading coefficient. The graphical approach reveals the function's overall shape, direction, intercepts, and location of the vertex, which can be pivotal for understanding complex relationships represented by the equation. By graphing our model of the U.S. debt, we visually follow the debt's growth over time and anticipate future values, which is incredibly useful for both educational and predictive purposes.

Mathematical Model Prediction

Mathematical models are fundamental in predicting future outcomes based on historical data. They establish relationships between variables and can show how changes in one might affect another. However, predictions assume that current trends continue unchanged, and in reality, many factors can alter these trends. In this scenario, the model predicts when the debt will reach \$20 trillion by setting \( D = 20 \) and then solving for \( t \). While this offers a quantitative prediction, such models should be utilized with an understanding of their assumptions and potential deviations from future realities.

Validity of Mathematical Models

The validity of mathematical models depends on their ability to accurately represent the real world and predict future events. Critical factors like the model's assumptions, the quality of past data, and potential for future change must be considered. In the case of the public debt model, while it might have fit past data well, its validity for future predictions is questionable, as it doesn't account for unexpected political, economic, or social changes. Therefore, while models are powerful tools, one must critically evaluate their practical limitations before using them for long-term predictions.