Question

Population The population \(P\) of the United States officially reached 300 million at about 7:46 A.M. E.S.T. on Tuesday, October 17,2006 . The table shows the U.S. populations (in millions) since 1900. (Source: U.S. Census Bureau)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1900 & 76 \\ \hline 1910 & 92 \\ \hline 1920 & 106 \\ \hline 1930 & 123 \\ \hline 1940 & 132 \\ \hline 1950 & 151 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1960 & 179 \\ \hline 1970 & 203 \\ \hline 1980 & 227 \\ \hline 1990 & 250 \\ \hline 2000 & 282 \\ \hline 2006 & 300 \\ \hline \end{array} $$(a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=0\) corresponding to 1900 . (b) Use the regression feature of a graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Graph the exponential model in base \(e\) with the scatter plot of the data. What appears to be happening to the relationship between the data points and the regression curve at \(t=100\) and \(t=106 ?\) (d) Use the regression feature of a graphing utility to find a logistic growth model for the data. Graph each model using the window settings shown below. Which model do you think will give more accurate predictions of the population well beyond \(2006 ?\)

Short answer

Several mathematical models can be derived from the provided U.S. population data. An exponential model can be obtained using regression techniques on graphing software, and this model can be rewritten in base \(e\). Besides, a logistic growth model can also be obtained and compared with the exponential model to evaluate their predictive power beyond the year 2006.

Step-by-step solution

Creating the Scatter Plot

Firstly, input the given data into your graphing utility. Set \(t = 0\) for the year 1900 and plot the rest of the data accordingly. This will give a visual representation of the population data over time.

Finding the Exponential Model

Then, use the regression feature of your graphing tool to find an exponential model for the data. This will usually involve selecting a regression type (exponential in this case) and then allowing the software to calculate the result.

Rewrite the Model in Base \(e\)

Upon obtaining the exponential model, you'll need to rewrite it in base \(e\). You can do this by employing the inverse property \(b = e^{\ln b}\). This operation will transform the model to its equivalent version in base \(e\).

Graphing the Exponential Model and the Scatter Plot Together

The next step is to graph the newly found exponential model with the scatter plot of the data. Pay attention to the relationship between the data points and the regression curve at \(t=100\) and \(t=106\). Record any specific patterns or anomalies you notice.

Finding the Logistic Growth Model

Next, use the regression feature of your graphing utility again, but this time, find a logistic growth model for the data. Also, graph this model.

Comparing the Models

Finally, compare both models (Exponential and Logistic) to make predictions about future population growth. An evaluation of how well each model fits the data will reveal which model you think will give a more accurate prediction for population growth beyond 2006. The choice is usually based on how well the model represents the given data and the properties inherent to the specific model (exponential or logistic).

Key concepts

Exponential Growth

Exponential growth happens when the increase in a quantity is proportional to the current amount, resulting in growth that accelerates over time. This is often represented by the formula \( P(t) = P_0 e^{rt} \), where \( P_0 \) is the initial amount, \( r \) is the growth rate, and \( t \) is time. Exponential growth is characterized by its potential for rapid increase.

In the context of population growth, an exponential model predicts that as the population increases, it grows more rapidly because there are more individuals reproducing. This model was used in our exercise to understand how the U.S. population has grown over time.

An important aspect of using exponential models is recognizing when the growth predicted may not be realistic indefinitely, especially as resources become limited or other factors come into play.

Logistic Growth

Logistic growth models are often more suitable for populations over the long term compared to exponential models. This is because logistic growth considers limits to growth such as competition for resources or social factors.

The logistic growth equation is: \[ P(t) = \frac{K}{1 + \frac{K - P_0}{P_0} e^{-rt}} \]Where \( K \) is the carrying capacity, or maximum population the environment can sustain, \( P_0 \) is the initial population size, and \( r \) is the growth rate.

In contrast to exponential growth, logistic growth results in a sigmoid curve which starts with exponential growth but slows as it approaches the carrying capacity. This makes logistic models a better choice for predicting long-term growth trends when constraints are involved.

Scatter Plot

A scatter plot is a powerful tool used to visualize relationships between two variables. In the context of this exercise, a scatter plot was used to display U.S. population data over time from 1900 to 2006. Placing time on the horizontal axis and population on the vertical axis, allows us to visually assess the trend and the form of the data distribution.

By looking at the scatter plot, we can begin to understand whether the data follows a particular pattern, such as linear, exponential, or logistic. It is the first step in identifying which mathematical model to apply to describe the data accurately.

Using scatter plots is foundational in data analysis, helping students and researchers make informed decisions about the appropriate statistical or mathematical models to apply for further analysis.

Regression Analysis

Regression analysis involves identifying relationships between variables and fitting a mathematical model to the data. This is a crucial step in making predictions based on historical data.

In the exercise, regression tools were employed to fit both exponential and logistic growth models to U.S. population data. Regression analysis helps compute the model parameters by minimizing errors between the model predictions and the actual data.

• Exponential Model: Fits data when exponential growth is expected. It shows how well the model predicts future trends if growth accelerates.
• Logistic Model: More relevant when a population is expected to approach a carrying capacity. It balances initial growth rates with limits on resources.

Using regression in this way allows us to quantify how closely each model fits the data, which is essential when deciding between different growth models. Understanding how well a model fits allows better forecasting and planning based on the analyzed trends.