Problem 78 - Contemporary Precalculus

Question

The total health care expenditures \(E\) in the United States (in billions of dollars) can be approximated by $$ E=73.04 x+625.6 $$ where \(x\) is the number of years since \(1990 .^{\dagger}\) Determine the year in which health care expenditures are at the given level: \(\$ 1940.3\) billion

Step-by-Step Solution

Answer

Answer: 2008

Step 1: Write the given equation

We are given the equation \(E = 73.04x + 625.6\), which describes the health care expenditures in billions of dollars.

Step 2: Substitute the given expenditure

Substitute the given health care expenditure, \(1940.3\) billion, into the equation: \(1940.3 = 73.04x + 625.6\)

Step 3: Solve for x

Isolate x by subtracting \(625.6\) from both sides and then divide by \(73.04\): \(1940.3 - 625.6 = 73.04x\) \(1314.7 = 73.04x\) \(x = \dfrac{1314.7}{73.04}\) \(x \approx 18\)

Step 4: Determine the year

The value of x represents the number of years since 1990. To find the year in which health care expenditures are at the given level, add \(x\) to \(1990\): \(1990 + 18 = 2008\) In conclusion, health care expenditures reach \(1940.3\) billion in the year \(2008\).

Key Concepts

Understanding Linear Equations

Linear equations are foundational in algebra and are characterized by each term being either a constant or the product of a constant and a single variable. In the context of health care expenditures, we used the linear equation E = 73.04x + 625.6 where E represents the total expenditures in billions of dollars and x is the number of years since 1990.

The formula given is in the form y = mx + b, where m is the slope of the line representing the rate of change in cost per year, and b is the y-intercept, which represents the initial cost in the baseline year of the study. By understanding this structure, students can easily insert known variables to predict unknowns within the context of the problem.

When the problem requires solving for the year corresponding to a specific expenditure, substitution of the given value for E allows us to solve for x, which then leads us to the particular year when such expenditure levels were expected to be reached, using basic algebraic manipulations.

Mathematical Modeling in Real-World Scenarios

Mathematical modeling involves creating a mathematical representation of a real-world scenario. It allows us to predict future events, analyze trends, and make decisions based on modeled forecasts. In our health care expenditure problem, the linear model is built upon historical data and the assumption that trends continue linearly.

The model E = 73.04x + 625.6 doesn't just stand alone but is derived from historical examination and statistical analysis of past health care expenditures. By using such models, we can make informed predictions about future costs, which is critical for budgeting and policy-making in healthcare.

However, it's important to note that mathematical models are simplifications of reality. They can be affected by fluctuations in economic conditions, changes in policy, and unforeseen events. Thus, while helpful, models should be viewed as approximates rather than exact predictions.

Problem-solving with Algebra

Problem-solving in algebra involves a logical sequence of steps to find the answer to a given problem. In the health care expenditure exercise, the steps were clear: identify the knowns and unknowns, substitute known values into the equation, and use algebraic principles to isolate the unknown variable.

This systematic approach is a valuable skill not only in mathematics but also in various real-life scenarios where logical thinking and methodical processes are necessary. The exercise demonstrates how algebra can provide clear solutions to problems that initially appear complex, breaking them down into manageable steps.

When faced with algebraic problems, it's important for students to:

• Visualize the problem and identify the variables involved.
• Construct an equation with the given information.
• Execute a series of algebraic steps to solve for the unknown.
• Interpret the solution and apply it to the context of the problem.

Employing these strategies encourages successful problem-solving and builds a strong foundation for tackling more challenging mathematical concepts.