Question

Financial Aid The average award for Federal Pell Grants and Federal Perkins Loans from 1995 to 2005 can be approximated by \(=-2.051 t^{3}+56.87 t^{2}-376.7 t+2238\) Federal Pell Grant \(=-1.810 t^{3}+56.64 t^{2}-476.4 t+2711\) Federal Perkins Loan where \(A\) is the award (in dollars) and \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use a graphing utility to determine whether Federal Perkins Loan awards will exceed Federal Pell Grant awards. Do you think these models will continue to be accurate? Explain your reasoning. (Source: U.S. Department of Education)

Short answer

The process of comparing the awards begins with plotting the two third degree polynomial functions for the Federal Pell Grant and Federal Perkins Loan in a graphing calculator. Where and if the Perkins Loan line exceeds the Pell Grant line, the Perkins Loan awards will exceed. Given these models are simple polynomial functions, they may not remain accurate in the future as they may not account for all influencing factors such as changes in governmental policies or economic situations.

Step-by-step solution

Understand the Variables and Equations

Identify the variables, interpret the equations provided for Federal Pell Grants \(A = -2.051 t^{3} + 56.87 t^{2} - 376.7 t + 2238 \) and Federal Perkins Loans \(A = -1.810 t^{3} + 56.64 t^{2} - 476.4 t + 2711\). Here, \(A\) is the award (in dollars) and \(t\) represents the year, with \(t=5\) corresponding to 1995.

Plot the Equations using a Graphing Utility

Use a graphing utility to plot both equations. This will allow you to visually compare the value of each grant. Since the problem doesn't specify the exact type of graphing utility, it can assumed any such tool (like online graphing calculators or software) can be used.

Compare the Models

Observe and analyze the plotted graphs closely to see which grant exceeds the other. Look specifically at the points where the two graphs intersect. If the Perkins Loan plot line rises above the Pell Grant line at any point, then there are times when the Perkins Loan awards exceed Pell Grant awards.

Consider the Future Accuracy of the Models

In considering whether the models will continue to be accurate, one should consider that the given models are polynomial and may not account for all the factors that can influence Federal Grants and Loans such as changing governmental policies, economic fluctuations, etc. Hence, there is a high likelihood that these models may not remain accurate in the future.

Key concepts

Federal Pell Grants

Federal Pell Grants are a form of financial aid provided by the federal government to help undergraduates with exceptional financial need. These grants are typically awarded on the basis of the student's financial situation, and unlike loans, do not need to be repaid.

The formula given in the exercise, \(A = -2.051 t^3 + 56.87 t^2 - 376.7 t + 2238\), estimates the average Federal Pell Grant award over a specified time period. In this polynomial equation, \(A\) represents the award amount in dollars, and \(t\) denotes the year, with a specific year (1995) associated with \(t=5\). The cubic polynomial model reflects different trends over time, such as increases or decreases in award amounts due to legislative changes, budget allocations, or adjustments for inflation.

Federal Perkins Loans

Federal Perkins Loans used to be low-interest federal student loans for undergraduate and graduate students with exceptional financial need. The repayment of these loans begins after a student graduates, leaves school, or drops below half-time enrollment.

The provided equation, \(A = -1.810 t^3 + 56.64 t^2 - 476.4 t + 2711\), models the average Federal Perkins Loan award, with \(A\) again representing award dollars and \(t\) the year, normalized so that \(t=5\) is the year 1995. Like the Pell Grants, this equation highlights trends in loan award amounts over the given period.

Polynomial Models in Economics

Polynomial models, such as those given for the financial aid awards, are widely used in economics to forecast variables over time. These models, including quadratic, cubic, or higher-degree polynomials, fit past data to predict future trends.

However, the accuracy of such models can diminish over time, as they are best-suited for short-term forecasting. Economic variables often have complex relationships with multiple influencing factors, and a polynomial model might not capture unexpected changes in policies, economic conditions, or other external factors. Thus, while useful, it is essential to continually reassess polynomial models against real-world data.

Graphing Utility in Algebra

Graphing utilities are essential tools in algebra, enabling users to visualize complex equations and their solutions. The graphical representation facilitates comparison of different functions, such as in our exercise where the awards for Federal Pell Grants and Federal Perkins Loans are plotted.

By inputting the respective polynomial equations into a graphing utility, we can easily see intersections, maxima, and minima, which are key features in understanding the behavior of the function over time. This visual approach makes it simpler to compare the magnitude of the grants over the years and predict when one might exceed the other.