Problem 54 - College Algebra

Question

The graph on the next page shows the number of Americans enrolled in HMOs, in millions, from 1992 through \(2000 .\) The data can be modeled by the exponential function $$f(x)=36.1 e^{0.113 x}$$ which describes enrollment in HMOs, \(f(x),\) in millions, \(x\) years after \(1992 .\) Use this function to solve. According to the model, how many Americans will be enrolled in HMOs in the year \(2008 ?\) Round to the nearest tenth of a million.

Step-by-Step Solution

Answer

According to the model, approximately 81.6 million Americans will be enrolled in HMOs in the year 2008.

Step 1: Identify the year in question relative to the base year

The function is given in terms of years after 1992, so the first step is to determine how many years after 1992 is 2008. Subtract 1992 from 2008 to get this value. Thus, \(x = 2008 - 1992 = 16\).

Step 2: Substitute the value of \(x\) into the function

The next step is to substitute the value of \(x = 16\) into the function to get the predicted enrollment for that year. Thus, \(f(16) = 36.1 e^{0.113*16}\).

Step 3: Calculate the predicted enrollment

The final step is to actually calculate the value of the function. As per the calculator, \(f(16) = 36.1 e^{0.113*16}\) approximately equals 81.6.

Key Concepts

HMO Enrollment Prediction

Understanding the future enrollment in Health Maintenance Organizations (HMOs) is crucial for healthcare planning and management. To predict this figure, an exponential function can be an effective tool. Essentially, it translates historical data into a mathematical model that estimates future growth. The exercise provides such a function specifically for HMO enrollments:
The formula \(f(x) = 36.1 e^{0.113 x}\) represents the number of millions of Americans enrolled in HMOs, with \(x\) being the number of years after 1992. To make a prediction for the year 2008, you simply input the respective year’s \(x\) value into the function.
Shorter sentences simplify comprehension: The base year, 1992, is our starting point. By 2008, 16 years have passed. Plug in 16 for \(x\) to get \(f(16)\). The calculation provides an enrollment prediction. Use a calculator for \(e^{0.113*16}\) then multiply by 36.1 to find the predicted enrollment, to the nearest tenth of a million.
This utilization of exponential functions for enrollment prediction aids in resource allocation and policy development within healthcare institutions.

Exponential Growth Modeling

When depicting growth that increases at a rate proportional to its current value, exponential functions are key. This kind of growth is common in populations, finance, and—as shown in the exercise—HMO enrollments.
An exponential function like \(f(x) = ab^{x}\), where \(a\) is the initial amount and \(b\) is the growth factor, can depict this rapidly increasing trend. For the HMO model, \(a = 36.1\) million represents the starting enrollment count, and \(b = e^{0.113}\) illustrates the growth rate per year.

Why Use Exponential Models?

• They capture growth that increases faster over time.
• They provide predictive insight for planning and strategy.
• They help understand complex phenomena in a comprehensible way.

Expressing data exponentially aids in anticipating future scenarios by reflecting past trends and projecting them forward in time. Teaching students how to use these models allows them to engage in advanced planning for various real-world applications.

Interpreting Exponential Functions in Algebra

Exponential functions are one of the fundamental tools in algebra for representing and analyzing scenarios where quantities grow or decay rapidly. The base of the exponent, in this case \(e\), known as Euler’s number, is crucial when the rate of change is continuous.
The given formula \(f(x) = 36.1 e^{0.113 x}\) provides a base for interpreting exponential functions. Here, \(36.1\) is the initial value when \(x = 0\), reflecting the enrollment in 1992. The exponent, \(0.113x\), indicates the growth rate. The larger \(x\) gets, the bigger the impact of the exponent on the total enrollment, showing exponential growth.

Breaking Down the Components

• \(\(36.1\)\): Initial value or the enrollment at the reference year (1992).
• \(\(e\)\): The base indicating continuous growth rate in this context.
• \(\(0.113\)\): The growth rate percentage expressed as a decimal.
• \(\(x\)\): The number of years after the base year, 1992 here.

To interpret exponential functions in an algebraic context, one must understand not just the components of the equation, but also the real-world concept it models—such as HMO enrollments' growth over time. Engaging with real-life examples, like this healthcare model, bolsters comprehension and application of exponential functions in algebra.