The table shows the total receipts R and total expenditures...

The table shows the total receipts $R$ and total expenditures $E$ for the Old-Age and Survivors Insurance Trust Fund (Social Security Trust Fund) in billions of dollars. The time $t$ is given in years, with $t=1$ corresponding to 1991 . $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \boldsymbol{t} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{R} & 299.3 & 311.2 & 323.3 & 328.3 & 342.8 & 363.7 \\ \hline \boldsymbol{E} & 245.6 & 259.9 & 273.1 & 284.1 & 297.8 & 308.2 \\ \hline \end{array} $$ $$ \begin{array}{|l|c|c|c|c|c|} \hline \boldsymbol{t} & 7 & 8 & 9 & 10 & 11 \\ \hline \boldsymbol{R} & 397.2 & 424.8 & 457.0 & 490.5 & 518.1 \\ \hline \boldsymbol{E} & 322.1 & 332.3 & 339.9 & 358.3 & 377.5 \\ \hline \end{array} $$ (a) Use a graphing utility to fit an exponential model to the data for receipts. Plot the data and graph the model. (b) Use a graphing utility to fit an exponential model to the data for expenditures. Plot the data and graph the model. (c) If the models are assumed to be true for the years 2002 through 2007, use integration to approximate the surplus revenue generated during those years. (d) Will the models found in parts (a) and (b) intersect? Explain. Based on your answer and news reports about the fund, will these models be accurate for long-term analysis?

The table shows the total receipts $R$ and total expenditures $E$ for the Old-Age and Survivors Insurance Trust Fund (Social Security Trust Fund) in billions of dollars. The time $t$ is given in years, with $t=1$ corresponding to 1991 .
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline \boldsymbol{t} & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline \boldsymbol{R} & 299.3 & 311.2 & 323.3 & 328.3 & 342.8 & 363.7 \\
\hline \boldsymbol{E} & 245.6 & 259.9 & 273.1 & 284.1 & 297.8 & 308.2 \\
\hline
\end{array}
$$
$$
\begin{array}{|l|c|c|c|c|c|}
\hline \boldsymbol{t} & 7 & 8 & 9 & 10 & 11 \\
\hline \boldsymbol{R} & 397.2 & 424.8 & 457.0 & 490.5 & 518.1 \\
\hline \boldsymbol{E} & 322.1 & 332.3 & 339.9 & 358.3 & 377.5 \\
\hline
\end{array}
$$
(a) Use a graphing utility to fit an exponential model to the data for receipts. Plot the data and graph the model.
(b) Use a graphing utility to fit an exponential model to the data for expenditures. Plot the data and graph the model.
(c) If the models are assumed to be true for the years 2002 through 2007, use integration to approximate the surplus revenue generated during those years.
(d) Will the models found in parts (a) and
(b) intersect? Explain. Based on your answer and news reports about the fund, will these models be accurate for long-term analysis?

Key Concepts

Numerical Integration for Accumulated Change

Numerical integration is the process of approximating the total accumulation of a quantity, such as net revenue, over a given interval by summing up small changes. In this context, it is used to approximate the surplus revenue by integrating the difference between two functions (receipts and expenditures) over a specified time period.

Intersection Analysis of Functions

Analyzing the intersection of functions involves finding if and when two modeled curves, such as those representing receipts and expenditures, cross each other. This concept helps in understanding at what point two quantities become equal, which is important for financial analysis and decision making.

Exponential Modeling

Exponential modeling involves representing data with an equation of the form y = ab^t, where the base b characterizes the rate of increase or decrease and t represents time. This concept is central to analyzing phenomena that exhibit consistent proportional change over time, such as financial growth, population dynamics, and decay processes.

Curve Fitting Using Graphing Utilities

Curve fitting using graphing utilities refers to the process of using tools like graphing calculators or software to obtain the best-fit parameters for a chosen model that describes a set of data. This technique is crucial in data analysis to determine the mathematical relationships between variables and to predict future behavior based on historical trends.

Model Extrapolation and Long-Term Validity

Extrapolation involves extending a model beyond the range of observed data to predict future outcomes. While useful for short-term forecasting, the long-term validity of models based on exponential trends can be limited due to changing external factors. This concept highlights the caution needed when applying fitted models over extended periods, especially in complex systems like financial scenarios.

Transcript

00:01 In this question we have a table in which t represents time, r represents receipts and e represents expenditure for the old age and survivors insurance trust fund.

00:14 We are required to fit an exponential model for the receipts with the help of graphing utility.

00:22 After that, we are required to fit an exponential model for the expenditure by using graphing utility.

00:29 After that, we are required to find the surplus revenue generated during 2002 to 2007.

00:38 And finally, we need to explain will the model a and b intersect and we need to decide will these models be accurate for long -term analysis.

00:49 So let's see how to solve this question.

00:51 Part a, when we use the graphing utility, the exponential model for the receipts can be written as y1.

00:59 Is equals to 270 .31 a to the power 0 .695 t and the plot or the graph for this model is shown below.

01:25 So this is the graph for this model.

01:29 Y1 is equal to 270 .3151 e to the power 0 .05695t.

01:40 And this is the final answer for part a.

01:43 And now let's move to part b.

01:48 Part b.

01:50 When we use the graphing utility, the exponential model for the expenditure can be written as y2 is equals to 239 .9 .9704 a to the power 0 .04074t and the graph or the plot for this exponential model is shown below...

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