Problem 12
Question
Park City Population (Historic) Park City, Utah was settled as a mining community in 1870 and experienced growth until the late \(1950 \mathrm{~s}\) when the price of silver dropped. In the past 40 years, Park City has experienced new growth as a thriving ski resort. The population data for selected years between 1900 and 2009 are given below. Park City, Utah \begin{tabular}{|c|c|} \hline Year & Population \\ \hline 1900 & 3759 \\ \hline 1930 & 4281 \\ \hline 1940 & 3739 \\ \hline 1950 & 2254 \\ \hline 1970 & 1193 \\ \hline 1980 & 2823 \\ \hline 1990 & 4468 \\ \hline 2000 & 7341 \\ \hline 2009 & 11983 \\ \hline \end{tabular} (Source: Riley Moffart, Popularion History of Western U.S. Citio \(d\) Towne, \(1850-1990\), Lanham: Scarecrow, 1996 , 309 ; and U.S. Bureat of the Census) a. What behavior of a scatter plot of the data indicates that a cubic model is appropriate? b. Align the input so that \(t=0\) in \(1900 .\) Find a cubic model for the data. c. Numerically estimate the derivative of the model in 2008 to the nearest hundred. d. Interpret the answer to part \(c\).
Step-by-Step Solution
Verified Answer
a: Non-linear trend with two increases and a dip. b: Adjust years to have 1900 as year 0 and fit cubic model. c, d: Derivative at 2008 gives population change rate.
1Step 1: Analyze the Data Trend
Create a scatter plot using the given population data over the years. Look for a non-linear pattern that might suggest the use of a cubic model. If the data points show an initial increase, a dip, and then another increase, a cubic model may be suitable.
2Step 2: Set the Origin for Time
Align the input so that the year 1900 corresponds to time zero (i.e., let \( t = 0 \) in 1900). This means we will subtract 1900 from each year to get the corresponding value of \( t \). For example, 1930 will become \( t = 30 \), 1940 will become \( t = 40 \), and so on.
3Step 3: Fit a Cubic Model
Use the adjusted time values and population figures to fit a cubic polynomial model: \( P(t) = at^3 + bt^2 + ct + d \). You can use technology, like a graphing calculator or statistical software, to determine the coefficients \(a\), \(b\), \(c\), and \(d\) by performing a regression fit on the data.
4Step 4: Calculate the Derivative of the Model
The derivative of the cubic model is \( P'(t) = 3at^2 + 2bt + c \). To find the rate of change of the population in 2008, calculate \( P'(108) \), as 2008 corresponds to \( t = 108 \) when \( t = 0 \) is 1900.
5Step 5: Interpret the Rate of Change
The derivative value calculated in Step 4 represents the rate of change of the population in 2008. This rate indicates how quickly the population was increasing or decreasing in that year, expressing how many people were added or removed per year.
Key Concepts
Scatter Plot AnalysisNumerical DerivativePopulation Trend InterpretationPolynomial RegressionHistorical Data Analysis
Scatter Plot Analysis
When analyzing population data, creating a scatter plot is crucial. A scatter plot provides a visual representation of how the data points spread across different years. For the Park City population data, each data point represents the population at a specific year.
Scatter plot analysis helps identify patterns within the data set. In particular, it reveals whether each data point follows a linear or non-linear pattern. In this scenario, a cubic model may be appropriate if the data points demonstrate a rise, followed by a decline, and then another rise. This behavior suggests a more complex relationship among the variables that cannot be captured by a simple linear model.
By plotting the data, you can visually determine whether a cubic polynomial regression is suitable for modeling population trends over time. The key is to observe the pattern the plot exhibits and match it to a possible mathematical model.
Scatter plot analysis helps identify patterns within the data set. In particular, it reveals whether each data point follows a linear or non-linear pattern. In this scenario, a cubic model may be appropriate if the data points demonstrate a rise, followed by a decline, and then another rise. This behavior suggests a more complex relationship among the variables that cannot be captured by a simple linear model.
By plotting the data, you can visually determine whether a cubic polynomial regression is suitable for modeling population trends over time. The key is to observe the pattern the plot exhibits and match it to a possible mathematical model.
Numerical Derivative
Understanding the rate of change in population over time involves computing the derivative of the population model. The derivative gives us the instantaneous rate of change, offering a snapshot of how the population is growing or shrinking at a particular moment.
For a cubic model represented by the equation \( P(t) = at^3 + bt^2 + ct + d \), its derivative is \( P'(t) = 3at^2 + 2bt + c \). This results in a quadratic equation, based on the coefficients from the polynomial.
To numerically estimate the derivative for a given year, plug in the corresponding \( t \) value. For instance, if you need the rate of change in 2008 and you've aligned the year 1900 as \( t = 0 \), then 2008 becomes \( t = 108 \). By substituting \( t = 108 \) into the derivative equation, you estimate the derivative, providing the rate of change in population for that year.
For a cubic model represented by the equation \( P(t) = at^3 + bt^2 + ct + d \), its derivative is \( P'(t) = 3at^2 + 2bt + c \). This results in a quadratic equation, based on the coefficients from the polynomial.
To numerically estimate the derivative for a given year, plug in the corresponding \( t \) value. For instance, if you need the rate of change in 2008 and you've aligned the year 1900 as \( t = 0 \), then 2008 becomes \( t = 108 \). By substituting \( t = 108 \) into the derivative equation, you estimate the derivative, providing the rate of change in population for that year.
Population Trend Interpretation
Population trend interpretation allows us to make sense of the numerical derivative and its implications. The rate of change, or derivative, tells us the slope at which the population is increasing or decreasing.
The calculated derivative for a given year quantifies the speed of population change. If the result is positive, the population is increasing. Conversely, a negative derivative indicates a declining population. The magnitude of the derivative reflects how rapidly these changes are occurring.
Interpretation goes beyond numbers to provide insight. For example, a high positive derivative in 2008 suggests significant growth, potentially due to Park City's development as a ski resort. Such interpretations help in planning and forecasting future population dynamics.
The calculated derivative for a given year quantifies the speed of population change. If the result is positive, the population is increasing. Conversely, a negative derivative indicates a declining population. The magnitude of the derivative reflects how rapidly these changes are occurring.
Interpretation goes beyond numbers to provide insight. For example, a high positive derivative in 2008 suggests significant growth, potentially due to Park City's development as a ski resort. Such interpretations help in planning and forecasting future population dynamics.
Polynomial Regression
Polynomial regression is a form of statistical modeling used to relate a dependent variable to one or more independent variables. For the Park City population data, polynomial regression involves fitting a cubic polynomial to the data points.
This method involves determining coefficients for the polynomial terms \( a \), \( b \), \( c \), and \( d \) in the model \( P(t) = at^3 + bt^2 + ct + d \). Various tools, such as graphing calculators or software like Excel, can assist with the regression analysis. These tools use numerical algorithms to identify the best-fit curve.
The result of polynomial regression is a mathematical model that approximates the historical population patterns, informing how past behaviors might project into future trends. It provides a more dynamic and accurate forecast compared to simple linear models.
This method involves determining coefficients for the polynomial terms \( a \), \( b \), \( c \), and \( d \) in the model \( P(t) = at^3 + bt^2 + ct + d \). Various tools, such as graphing calculators or software like Excel, can assist with the regression analysis. These tools use numerical algorithms to identify the best-fit curve.
The result of polynomial regression is a mathematical model that approximates the historical population patterns, informing how past behaviors might project into future trends. It provides a more dynamic and accurate forecast compared to simple linear models.
Historical Data Analysis
Analyzing historical data involves examining past records to uncover patterns and trends. For Park City, historical population data from 1900 to 2009 showcases significant shifts due to factors such as economic changes and urban development.
This form of data analysis uses past data points to predict future scenarios. By understanding historical context, such as the impact of mining or the development of the ski industry, one can better interpret changes in population data.
Historical data analysis plays a vital role in constructing predictive models. It enhances understanding of factors influencing population changes over time, proving essential for making informed decisions in urban planning and development.
This form of data analysis uses past data points to predict future scenarios. By understanding historical context, such as the impact of mining or the development of the ski industry, one can better interpret changes in population data.
Historical data analysis plays a vital role in constructing predictive models. It enhances understanding of factors influencing population changes over time, proving essential for making informed decisions in urban planning and development.
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