Question

Modeling Data The table shows the resident populations \(P\) (in millions) of the United States from 1920 to 2010 . (Source: U.S. Census Bureau) $$ \begin{array}{|c|c|c|c|c|}\hline \text { Year } & {1920} & {1930} & {1950} & {1960} \\ \hline \text { Population, } P & {106} & {123} & {132} & {151} & {179} \\ \hline \text { Year } & {1970} & {1980} & {1990} & {2000} & {2010} \\\ \hline \text { Population, } P & {203} & {227} & {249} & {281} & {309} \\\ \hline\end{array} $$ (a) Use the 1920 and 1930 data to find an exponential model \(P_{1}\) for the data. Let \(t=0\) represent 1920 . (b) Use a graphing utility to find an exponential model \(P_{2}\) for all the data. Let \(t=0\) represent 1920 . (c) Use a graphing utility to plot the data and graph models \(P_{1}\) and \(P_{2}\) in the same viewing window. Compare the actual data with the predictions. Which model better fits the data? (d) Use the model chosen in part (c) to estimate when the resident population will be 400 million.

Short answer

Substeps must be followed in order to answer this question completely as it involves several calculations, critical analysis and interpretation of data models for which the actual values will depend on the computations and graphing tool used. The estimation time for when the population will hit 400 million is particularly calculated depending on the selected model \(P_1\) or \(P_2\) from step 3.

Step-by-step solution

- Calculate Model P1

An exponent model has the form \(P(t) = P_0 \times e^{kt}\), where \(P(t)\) is the population at time \(t\) (year), \(P_0\) is the initial population, \(k\) is the constant rate, and \(e\) is Euler's number. To start, using the data from 1920 (t=0) and 1930 (t=10), we get two equations, \(106 = P_0 \times e^{0 \times k}\) (1) and \(123 = P_0 \times e^{10 \times k}\) (2). From equation 1, we can deduce that \(P_0 = 106\). Substituting \(P_0\) into equation (2), we can solve for \(k\) by dividing both sides by 106 and then take the natural log of both sides: \(ln(123/106)/10 = k\). Hence we can calculate \(k\) and have the exponent model \(P_1(t) = 106 \times e^{k \times t}\)

- Calculate Model P2

To find the exponential model \(P_2\), a graphing calculator or software that can apply a regression analysis to all data points given is required. Enter all the given data into the software and obtain the best fit exponential model, \(P_2(t)\). This requires a graphing calculator or software that has an exponential regression feature.

- Compare and Select Models

To choose the best model, plot all data points along with both models \(P_1\) and \(P_2\), using the same timeframe (1920 to 2010 = 0 to 90). The model that aligns properly with the data points is the better option. Visually inspect the deviation from each data point to the models to make a decision on which model is better.

- Predict Future Population

To determine the year with a population of 400 million, you would use the model determined in step 3, set \(P(t)\) equal to 400 and solve for \(t\). Then add the value of \(t\) to 1920 to get the estimated year. Note, you may also require the use of a logarithm to solve for \(t\) in this step depending on the model selected.

Key concepts

U.S. Census Data

The U.S. Census Data provides us with an official count and demographic makeup of the population in the United States. This data has been collected every ten years since 1790. For our exercise, we are particularly focusing on the census data from 1920 to 2010.
The U.S. Census Bureau collected the population numbers, which are vital in understanding the growth of the country's resident population over the decades.
Why is this data important? It allows us to model the population's growth trends, giving insights into urban planning, resource allocation, and predicting future population sizes.

• The stable intervals of 10 years help in providing consistent data points for modeling.
• It acts as a historical record showing growth and other demographic changes over time.

Using such data, we can develop models to make predictions about future population growth, which is particularly significant in how we plan for the future.

Graphing Utility

To effectively visualize and analyze our population growth models, we use a graphing utility. This tool is essential in displaying the growth of the population over time and helps in finding patterns or trends that may not be apparent through raw data.
Graphing utilities can range from simple calculators with graph capabilities to complex software like Excel or specialized statistical software. They are essential for:

• Plotting raw data points to understand trends visually.
• Overlaying various models, such as our exponential models, with the actual data for comparison.
• Helping in verifying the accuracy of different models by visual inspection.

In our exercise, a graphing utility aids in plotting the census data alongside the exponential models, allowing us to see which model aligns best with the real data.

Regression Analysis

Regression analysis is a statistical method used to estimate the relationships among variables. In our case, it is used to find the best-fit exponential model for the U.S. Census data.
When applying regression analysis to our data, we aim to develop an equation that best describes the underlying relationship between the time and the population growth. Here are some of the key aspects:

• It simplifies complex data points into a single line or curve that captures the trend.
• Reduces variability in large datasets by providing an average trend.
• It is used to predict future population sizes based on past data trends.

For the exponential growth model, the regression tool fits the data to the form of the equation using the least squares method to find the most accurate values for the model parameters.

Euler's Number

Euler's Number, denoted as \( e \), is approximately equal to 2.71828. This mathematical constant is crucial when dealing with exponential models. In the formula \( P(t) = P_0 \times e^{kt} \), \( e \) is the base of the natural logarithm.
Euler's Number is prevalent in scenarios involving continuous growth, such as population models or compound interest. Here's why it's indispensable:

• It provides a mathematical way to represent continuous growth rates.
• The natural logarithm, having base \( e \), simplifies many mathematical computations.
• Provides a standardized approach to modeling growth phenomena across different fields.

In the context of population modeling, \( e \) enables us to more easily calculate the rate at which the population increases or decreases over time, which is why it features in the exponential model formulas.