Question

Health Care Spending The total health care spending \(H\) (in millions of dollars) in the United States from 1995 to 2005 can be modeled by \(H=6136.36 t^{2}-22,172.7 t+979,909, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. The population \(P\) (in millions) of the United States from 1995 to 2005 can be modeled by \(P=3.0195 t+251.817, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. (Sources: U.S. Centers for Medicare and Medicaid Services and the U.S. Census Bureau) (a) Construct a rational function \(S\) to describe the per capita health spending. (b) Use a graphing utility to graph the rational function \(S\). (c) Use the model to predict the per capita health care spending in 2010 .

Short answer

The rational function to represent the per capita health spending is \(S(t) = \frac{6136.36 t^{2}-22,172.7 t+979,909}{3.0195 t+251.817}\). To graph this function, use a range from \(t=5\) to \(t=15\). The per capita health care spending prediction for 2010 can be found by inserting \(t=20\) into the rational function \(S(t)\). The exact numerical prediction would require actual computation.

Step-by-step solution

Construct the Rational Function S

To describe the per capita health spending, we need to create a rational function, S, defined as the health care spending (H) divided by the population (P). So, \(S = \frac{H}{P}\). Thus, plug the two models given in the exercise into this equation to get the new function \(S\), which represents per capita health care spending. So, \(S(t) = \frac{6136.36 t^{2}-22,172.7 t+979,909}{3.0195 t+251.817}\).

Graph the Rational Function S

Given that this is a written format, we cannot graph the function here. However, using a graphing utility such as Desmos or a graphing calculator, we can plot \(t\) in the x-axis ranging from 5 to 15 (representing the years from 1995 to 2005) and the values of \(S(t)\) in the y-axis.

Predict the Per Capita Health Care Spending in 2010

To predict the per capita health care spending in 2010, replace \(t\) in the function \(S(t)\) with the corresponding year. With \(t=5\) representing 1995, 2010 would correspond to \(t=20\). Therefore, compute \(S(20)\).

Key concepts

Rational Function

A rational function is a type of function commonly found in algebra and higher mathematics, represented by the ratio of two polynomial functions. Put simply, it’s just a fraction whereby both the numerator (the top part of the fraction) and the denominator (the bottom part) are polynomials. An important characteristic of a rational function is that it can have points where it is not defined, usually where the denominator is equal to zero.

For instance, in the context of the healthcare spending problem, the rational function is used to describe per capita health spending. This is the total health care spending divided by the population. Mathematically, it's written as:
\( S(t) = \frac{H(t)}{P(t)} \)
where \(H(t)\) and \(P(t)\) are polynomial functions that model health care spending and population, respectively. This function gives us valuable insights into how spending patterns change relative to population size over time. Using rational functions for modeling can highlight trends and is essential for predictive analytics in various sectors, including economics.

Health Care Spending Model

The health care spending model is an equation used to represent the amount of money spent on health care services over a period of time. In the given problem, the health care spending in the United States is modeled by a quadratic function:
\(H = 6136.36t^2 - 22,172.7t + 979,909\text{ for } 5 \leq t \leq 15\)

Within the exercise, \(t\) represents the year since 1995, starting with \(t=5\) in 1995. Quadratic models such as this are particularly useful because they can show how spending accelerates or decelerates over time. The coefficients (numbers in front of the \(t\)) tell us about the magnitude and direction of change. In this case, the model predicts that health care spending has been increasing as the years progress.

Understanding how to interpret these models helps in visualizing long-term financial trends, which is crucial for planning and decision-making in the health care sector.

Population Growth Model

When studying demographics, a population growth model is vital for understanding changes in population size over time. In the textbook exercise, the linear model:
\(P = 3.0195t + 251.817\text{ for } 5 \leq t \leq 15\)

represents the population of the United States, with \(t\) being the number of years since 1995. This linear model assumes a steady growth rate. The slope, 3.0195, suggests a constant increase in the population every year. Unlike quadratic models, which can represent accelerating growth, a linear model like this assumes that growth does not change rate; it's consistent.

It's crucial for students to appreciate these differences, as they influence how we might project future populations and thus affect planning in areas like urban development, resource allocation, and of course, health care.

Graphing Utility

A graphing utility, such as Desmos or a graphing calculator, is a powerful tool for visualizing functions and their behaviors. Whether digital or hand-held, these utilities allow for the plotting of complex equations, offering a visual representation that aids in comprehension and analysis of functions.

In educational settings, students are often asked to use these utilities to examine the slope, intercepts, concavity, and any asymptotes or discontinuities of functions. In the health spending problem, students can utilize these tools to plot the rational function \(S(t)\) and observe how per capita health spending evolves over time. By doing so, they can gain insights into the relationship between health care spending and population growth—themes critical to data-driven policy-making in health economics. These utilities are indispensable in modern math education and are particularly effective in helping students transform abstract concepts into tangible, understandable graphs.