Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

The average salaries \(S\) (in thousands of dollars) of secondary classroom teachers in the United States from 2005 through 2013 can be approximated by the model $$S=-0.143 t^{2}+3.73 t+32.5,5 \leq t \leq 13$$ where \(t\) represents the year, with \(t=5\) corresponding to \(2005 .\) (a) Determine algebraically when the average salary of a secondary classroom teacher was \(\$ 50,000\). (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 determine when the average salary reached \(\$ 55,500\) (e) Do you believe the model could be used to predict the average salaries for years beyond \(2013 ?\) Explain your reasoning.

Step-by-Step Solution

Answer

The solution to part (a) and (d) will be the values of \(t\) for which \(S = 50\) and \(S=55.5\) respectively. The table of values from step 2 will affirm or disprove the result from part (a) while the graph in part (c) will show a general trend of the average salary over the years 2005 to 2013. The last part will result in an explanation depending on critical interpretation of the problem model.

Step 1: Solve for \(t\) when \(S = 50\)

Substitute \(S = 50\) into the equation to determine when the average salary was \$50,000. Solve the equation \(-0.143 t^{2}+3.73 t + 32.5 = 50\). Simplify this equation to \(-0.143 t^{2}+3.73 t - 17.5 = 0\) and use the quadratic formula to find the values of \(t\). The quadratic formula is \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a=-0.143\), \(b=3.73\), and \(c=-17.5\).

Step 2: Create a table of values for the model

Using the equation \(S=-0.143 t^{2}+3.73 t+32.5\), compute the values of \(S\) for the range \(5 \leq t \leq 13\) and display these in a table. Include the obtained values of \(t\) from step 1 in the table to verify your results.

Step 3: Graph the model

Use the equation \(S=-0.143 t^{2}+3.73 t+32.5\) and the table of values created in the previous step to graph the model using any graphing utility.

Step 4: Determine when \(S = 55.5\)

Substitute \(S = 55.5\) into the function to determine when the average salary will reach \$55,500. Solve the equation \(-0.143 t^{2}+3.73 t + 32.5 = 55.5\). Simplify this equation to \(-0.143 t^{2}+3.73 t - 23 = 0\) and use the quadratic formula (as in step 1) to find the values of \(t\).

Step 5: Determine the usability of the model for years beyond 2013

Generally, mathematical models, including this one, can only predict within a valid range. Deciding whether this model could be useful to predict average salaries beyond 2013 will involve analysing the trends in the graph and previous years salaries, the nature of the problem, and the characteristics of the quadratic function itself.

Key Concepts

Average Salaries

The concept of average salaries is central to many economic and financial discussions. Average salaries represent the mean income obtained by individuals in a specific occupation, providing insight into income distribution and living standards. Here, we focus on the average salaries of secondary classroom teachers from 2005 to 2013, using a quadratic function as a model. It's important to remember that average salaries can reflect various factors:

• Experience level of employees
• Location of the employment
• Availability of resources and funding

In this exercise, the average salary is given in thousands of dollars and expressed as a function of time. The function allows us to estimate salary changes over the years, making it a powerful tool for understanding how these averages evolved within the given timeframe.

Graphing Models

Graphing models give a visual representation of relationships between variables, facilitating easy interpretation of trends over time. For our study of average salaries, we graph the equation \[ S = -0.143 t^{2} + 3.73 t + 32.5 \]where \(t\) stands for the years since 2005. By plotting this on a graph, we can clearly see how salaries progress throughout these years. Follow these steps to effectively graph the model:

• Choose an appropriate range for the variable \(t\), here between 5 and 13, representing the years 2005 to 2013.
• Plot the function at various points (e.g., \(t = 5, 6, 7, \ldots , 13\)).
• Connect these points to observe the trend over time.

The resulting curve will typically be parabolic, reflecting the quadratic nature of the equation.

Algebraic Equations

Algebraic equations serve as the backbone of mathematical modeling. They help to express relationships between different entities in a concise mathematical form. In this instance, the equation \[ S = -0.143 t^{2} + 3.73 t + 32.5 \]is used to calculate the average salaries of teachers and solve specific problems regarding salary thresholds.To determine when salaries reach a particular amount:

• Substitute the salary value into the equation (e.g., find \(t\) when \(S = 50\)).
• Rearrange the equation to a standard quadratic form \(at^2 + bt + c = 0\).
• Utilize the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), solving for \(t\).

These steps yield a timeline for when specific salary milestones are achieved.

Mathematical Modeling

Mathematical modeling is a crucial aspect of synthesizing real-world data into understandable and usable forms. This model of average teacher salaries over time is precisely that—a means to translate fluctuations in salaries into a manageable format. By using:

• A quadratic function, we capture both linear trends and curvature, representing acceleration or deceleration in salary changes.
• Constant variables (coefficients) allow the prediction of unknown values within the data range.

Modeling permits forecast and predictions – however, it is also limited to the constraints of available data. For years beyond 2013, careful consideration is needed. Factors like changing educational policies or economic conditions can affect the model's accuracy as they alter the underlying assumptions of constant coefficients and parameters. Therefore, while models provide valuable insights, they must be used judiciously for future predictions.