Problem 60 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Health-care spending per person (in dollars) by the private sector includes payments by individuals, corporations, and their insurance companies and is approximated by $$ 2.5 t^{2}+18.5 t+509 \quad(0 \leq t \leq 6) $$ where \(t\) is measured in years and \(t=0\) corresponds to the beginning of 1994 . The corresponding government spending (in dollars), including expenditures for Medicaid and other federal, state, and local government public health care, is $$ -1.1 t^{2}+29.1 t+429 \quad(0 \leq t \leq 6) $$ where \(t\) has the same meaning as before. Find an expression for the difference between private and government expenditures per person at any time \(t .\) What was the difference between private and government expenditures per person at the beginning of 1998 ? At the beginning of 2000 ?

Step-by-Step Solution

Answer

The difference between private and government expenditures at any time \(t\) is given by the function \(D(t) = 3.6t^2 - 10.6t + 80\). At the beginning of 1998, the difference between private and government expenditures per person was $95.2. At the beginning of 2000, the difference was $146.

Step 1: Find the difference between private and government expenditures function

To find an expression for the difference between private and government expenditures at any time \(t\), we need to subtract the government expenditure function from the private expenditure function. Let the difference function be \(D(t)\). Then, \(D(t) = P(t) - G(t)\) Substitute the given functions into the equation: \(D(t) = (2.5t^2 + 18.5t + 509) - (-1.1t^2 + 29.1t + 429)\) Now, simplify the expression: \(D(t) = 2.5t^2 + 18.5t + 509 + 1.1t^2 - 29.1t - 429\) \(D(t) = (2.5 + 1.1)t^2 + (18.5 - 29.1)t + (509 - 429)\) \(D(t) = 3.6t^2 - 10.6t + 80\) So the difference between private and government expenditures at any time \(t\) is given by: \(D(t) = 3.6t^2 - 10.6t + 80\)

Step 2: Find the difference at the beginning of 1998

To find the difference at the beginning of 1998, which is 4 years after the beginning of 1994, we need to substitute \(t = 4\) into the function \(D(t)\): \(D(4) = 3.6(4)^2 - 10.6(4) + 80\) Calculate the result: \(D(4) = 3.6(16) - 42.4 + 80\) \(D(4) = 57.6 + 37.6\) \(D(4) = 95.2\) So, the difference between private and government expenditures per person at the beginning of 1998 is $95.2.

Step 3: Find the difference at the beginning of 2000

To find the difference at the beginning of 2000, which is 6 years after the beginning of 1994, we need to substitute \(t = 6\) into the function \(D(t)\): \(D(6) = 3.6(6)^2 - 10.6(6) + 80\) Calculate the result: \(D(6) = 3.6(36) - 63.6 + 80\) \(D(6) = 129.6 + 16.4\) \(D(6) = 146\) So, the difference between private and government expenditures per person at the beginning of 2000 is $146.

Key Concepts

Health-care expenditures

Health-care expenditures can be divided into spending by the private sector and spending by the government. Each of these has its own unique factors influencing their amounts. The private sector includes spending by individuals, corporations, and insurance companies. Meanwhile, the government spending comprises expenditures for public services, including Medicaid and other federal, state, and local government public health care programs.
Understanding these expenditures helps us analyze how resources are being allocated and identify trends over time. The rates of these expenditures are often represented using mathematical functions, which allow for the calculation and prediction of spending levels at different times. These functions can help in planning budgets and policies within the health care sector.

Quadratic functions

Quadratic functions are mathematical equations that include terms up to the square of the variable. They take the form of: \[ ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. In the context of the given exercise, quadratic functions approximate the health-care expenditures for both the private and government sectors.
Quadratic functions are important because they can model real-world scenarios where changes accelerate over time, such as the growing health-care costs. They are depicted as parabolas when graphed, with the vertex representing either a maximum or minimum point. Understanding the components of the quadratic equation can help interpret how different elements of the expenditure may impact the overall cost. For instance, the coefficient of the \(t^2\) term affects the curvature of the graph, which in this exercise, represents the rate at which expenditures change overtime.

Difference calculations

Difference calculations involve finding the change between two quantities by subtracting one from the other. In applied mathematics, this is essential for analyzing how two functions compare across various points. For this exercise, we calculated the difference between private and government health-care expenditures. This gives insight into where more resources are being spent at given times, and by how much.
To find the difference, we subtracted the government expenditure function from the private expenditure function. This resulted in a new quadratic function that represents the difference in spending at any year \(t\). It's important to note that by evaluating this difference function at specific points, we can find exact discrepancies in expenditures over specific years, such as 1998 or 2000. This shows how much more was spent by the private sector compared to the government, or vice versa, and can aid in making informed policy decisions.

• Use the difference function for analyzing shifts in spending dynamics.
• Understanding these differences helps design better health-care budgets and policies.