Question

The total amount I (in millions of dollars) of healthcare expenditures and the residential population P (in millions) in the United States can be modeled by $$ \begin{aligned} &I=\frac{171,000 t+1,361,000}{1+0.018 t} \text { and } \\ &P=2.96 t+278.649 \end{aligned} $$ where \(t\) is the number of years since 2000. Find a model \(M\) for the annual healthcare expenditures per resident. Estimate the annual healthcare expenditures per resident in 2010.

Short answer

The model for the annual healthcare expenditures per resident is \(M= \frac{171,000 t^{2}+4.884 t+1,361,000 t+5,032,812.02}{2.96 t^{2}+0.05328 t^{2}+2.96 t+502.4642}\) in millions of dollars . The estimated annual healthcare expenditures per resident in 2010 is obtained by substituting \(t = 10\) into the above equation.

Step-by-step solution

Determining the model for M

Firstly, you need a model for the annual healthcare expenditures per resident, denoted as \(M\). This would be given by the ratio of healthcare expenditures to the population. So, \(M= \frac{I}{P}\). Substitute the given models for I and P into the formula for M to find the model for M.

Simplifying the expression

After substitification, you'll get \(M= \frac{\frac{171,000 t+1,361,000}{1+0.018 t}}{2.96 t+278.649}\). In order to simplify this complex fraction, multiply the numerator and denominator of the complex fraction by \((1+0.018 t)(2.96 t+278.649)\). The result is \(M= \frac{171,000 t^{2}+4.884 t+1,361,000 t+5,032,812.02}{2.96 t^{2}+0.05328 t^{2}+2.96 t+502.4642}\).

Estimating the healthcare expenditures per resident in 2010

For determining the healthcare expenditures per resident in 2010, plug \(t = 10\) into the M equation since the value of t represents the number of years since 2000. The resulting value represents the estimated annual healthcare expenditures per resident in 2010.

Key concepts

Rational Expressions

Rational expressions are mathematical expressions that resemble fractions, where both the numerator and the denominator are polynomials. In this problem, the expressions for healthcare expenditures \( (I) \) and the population \( (P) \) are examples of rational expressions. They involve variables combined through addition, subtraction, multiplication, and division.
Rational expressions are useful for modeling real-world scenarios where relationships between quantities are not simple or direct. By using rational expressions, we can easily describe and analyze these relationships, simplifying the complexity of the data into manageable formulas.
When working with rational expressions, it is crucial to understand the rules of algebra to perform operations like addition, subtraction, multiplication, and division. These operations help in simplifying models and solving equations, which is necessary for finding the model \( M \) in our exercise.

Fraction Simplification

Fraction simplification is an essential skill in algebra that involves reducing fractions to their simplest form. This process helps make complex expressions more manageable and easier to understand. In the given problem, we started with a complex fraction \( M = \frac{I}{P} \), where \( I \) and \( P \) are our rational expressions.
To simplify, substitute the values of \( I \) and \( P \) into \( M \) and deal with the resulting expression. This often involves breaking down polynomials, canceling common terms, and simplifying the resulting expression to make calculations easier.
Remember, the goal is to have a clear and concise expression. When simplification is done correctly, it leads to more straightforward calculations and more accessible interpretations of real-world problems. Here, it allowed us to compute the healthcare expenditures per resident.

Population Models

Population models are mathematical constructs used to estimate and predict population growth or changes over time. They often use linear, exponential, or even more complex mathematical expressions to represent how a population evolves. In our case, the population \( P \) is modeled by a linear expression \( P = 2.96t + 278.649 \).
These models are significant as they can provide insights into various fields like healthcare, economics, and environmental studies. For instance, understanding how the population changes can help governments plan future healthcare spending, social services, and infrastructural needs.
Accuracy in a population model is crucial as it strongly impacts the results of any related calculation, like determining per capita statistics, which are necessary for understanding economic and healthcare patterns.

Real-world Applications

Mathematical models are invaluable tools in understanding and solving real-world problems. They take complex data and processes, like healthcare expenditures and population data, and turn them into understandable mathematical expressions.
In practical scenarios, such as estimating healthcare expenditures per resident, mathematical models help predict future expenses and inform budget allocations and policy-making. By using the constructed model \( M \) for per capita expenditures, we gain insight into how the healthcare costs are spread across the population and can plan accordingly.
These real-world applications of algebra demonstrate the power and utility of mathematical concepts in addressing everyday issues, making planning and strategic decision-making more effective and precise.