Problem 53 - 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 \(2006 ?\) Round to the nearest tenth of a million.

Step-by-Step Solution

Answer

The model would predict that around 72.4 million Americans would be enrolled in HMOs in the year 2006.

Step 1: Understand the given function

Here, \(f(x) = 36.1e^{0.113x}\) is given as an exponential function to model the number of Americans enrolled in HMOs \(x\) years after 1992.

Step 2: Calculate x-value

We are asked to predict the number for the year 2006. Because the function runs off years after 1992, to find the x-value for 2006, we subtract 1992 from 2006 which gives us \(x = 14\).

Step 3: Substitute x value into the function

Now, with \(x = 14\), we plug this value into the equation: \(f(14) = 36.1e^{0.113*14}\).

Step 4: Calculate the function

By calculating the expression \(f(14) = 36.1e^{0.113*14}\), we find the predicted number of enrollments in millions.

Key Concepts

HMO enrollment modeling

Health Maintenance Organizations (HMOs) play an essential role in the healthcare system by offering a wide range of health services. Modeling their enrollment helps us understand trends and make predictions about future healthcare needs. In this exercise, the number of Americans enrolled in HMOs from 1992 onwards is modeled using an exponential function.Exponential growth is a key concept in this scenario. It indicates a rapid increase in enrollment over time, often seen in healthcare as demand and population increase. The exponential function provided, \(f(x) = 36.1e^{0.113x}\), represents this growth pattern. Here, \(f(x)\) estimates the number of enrollees in millions, with \(x\) reflecting the years after 1992. This model enables forecasting enrollments, thus helping stakeholders plan resource allocation, policy-making, and budgeting.

exponential function calculation

Calculating the exponential function accurately is vital for making correct predictions. The given function, \(f(x) = 36.1e^{0.113x}\), is designed to assess HMO enrollments over time. Let's break down the calculation steps to understand how to use this function.- **Identify the Base Year**: The exercise specifies 1992 as the base year. All calculations reference this starting point.- **Determine the Year of Interest**: For this problem, we are interested in the year 2006. Calculate \(x\), the number of years passed since 1992, by subtracting 1992 from 2006, resulting in \(x = 14\).- **Substitute \(x\) in the Function**: With the \(x\)-value determined, input \(x = 14\) into the function to get \(f(14) = 36.1e^{0.113\times14}\).- **Evaluate the Expression**: Using a calculator, compute the value of the expression \(e^{0.113\times14}\) and then multiply by 36.1 to find the HMO enrollment in millions for the specified year.

1992 to 2006 prediction

The years 1992 to 2006 mark a crucial period for evaluating HMO enrollment trends. This timeframe allows us to understand the pattern of exponential growth and make informed predictions.By examining this model, we can estimate the number of Americans likely enrolled in HMOs in 2006. The exponential model provides a straightforward way to do this. After performing the calculations, we substitute \(x = 14\) into \(f(x)\) and determine that the result, rounded to the nearest tenth, is the predicted enrollment in millions.This approach not only forecasts HMO trends over the specified years but also emphasizes the importance of exponential functions in making long-term predictions in healthcare. The ability to model such growth is invaluable for policymakers, healthcare providers, and economists to anticipate and prepare for future demands efficiently.