Problem 34
Question
The population \(P\) (in thousands) of Japan can be modeled by \(P=-14.71 t^{2}+785.5 t+117,216\) where \(t\) is time in years, with \(t=0\) corresponding to 1980 . (a) Evaluate \(P\) for \(t=0,10,15,20\), and 25 . Explain these values. (b) Determine the population growth rate, \(d P / d t\). (c) Evaluate \(d P / d t\) for the same values as in part (a). Explain your results.
Step-by-Step Solution
Verified Answer
The short answer would provide the numerical values for population \(P\) and the rate of population change \(dP/dt\) for each respective \(t\) and briefly summarize the trends in the population over the years.
1Step 1: Evaluating Population for Different Timelines
To find the population for given years, we substitute \(t=0\), \(t=10\), \(t=15\), \(t=20\), and \(t=25\) in given population function. For instance, when \(t=0\), \(P=-14.71*(0)^{2}+785.5*(0)+117,216 = 117,216\) thousand. Repeat this process for all given \(t\) values.
2Step 2: Finding the Population Growth Rate
The population growth rate is given by the derivative of the population model with respect to time, \(t\). This is written as \(dP/dt\). Differentiate \(P=-14.71 t^{2}+785.5 t+117,216\) with respect to \(t\) to find this rate.
3Step 3: Evaluating Population Growth Rate for Different Timelines
Substitute the \(t\) values from step 1 into the derivative obtained from step 2 to evaluate \(dP/dt\) for those points. For example, when \(t=0\) in the derivative, you will get the rate of population change for the year 1980.
4Step 4: Interpretation
Explain your results. Bear in mind that if the population growth rate \(dP/dt\) is positive, the population is increasing at that particular time. If \(dP/dt\) is negative, the population is decreasing, and if \(dP/dt = 0\), the population is constant with no change.
Key Concepts
Population Growth RateDerivative ApplicationsCalculus in DemographyMathematical Modeling
Population Growth Rate
Understanding the population growth rate is crucial when studying demographics and social sciences. It not only indicates changes over time but also helps predict future trends. The rate at which a population grows or shrinks can be represented mathematically and is fundamentally the first derivative of a population model with respect to time.
In the context of Japan's population model, the rate at which population changes, indicated by \( dP/dt \), helps us understand specific dynamics across different points in time. By taking the derivative of the polynomial function that models Japan's population, we obtain a new function that describes the rate of change in the population size for any given year since 1980.
Once we have this function, evaluating \( dP/dt \) for different years, such as in the textbook exercise, tells us the speed at which the population is increasing or decreasing at those particular times. This information can then guide policy makers to address demographic challenges.
In the context of Japan's population model, the rate at which population changes, indicated by \( dP/dt \), helps us understand specific dynamics across different points in time. By taking the derivative of the polynomial function that models Japan's population, we obtain a new function that describes the rate of change in the population size for any given year since 1980.
Once we have this function, evaluating \( dP/dt \) for different years, such as in the textbook exercise, tells us the speed at which the population is increasing or decreasing at those particular times. This information can then guide policy makers to address demographic challenges.
Derivative Applications
In calculus, derivatives serve as a powerful tool for understanding change, and their applications extend far beyond the classroom. The derivative of a function represents the rate of change or the slope of the function at any given point.
In the case of our textbook problem, applying derivatives to the population model gives insights into how population changes with time. This application of the derivative, known as the population growth rate, provides a snapshot of a country's demographic dynamics at different times. For policymakers and researchers, derivatives thereby become a critical component in understanding and responding to population trends.
In the case of our textbook problem, applying derivatives to the population model gives insights into how population changes with time. This application of the derivative, known as the population growth rate, provides a snapshot of a country's demographic dynamics at different times. For policymakers and researchers, derivatives thereby become a critical component in understanding and responding to population trends.
Calculus in Demography
Calculus plays a pivotal role in demography, the statistical study of populations. Through differentiation and integration, calculus provides techniques for modeling population trends, analyzing growth rates, and interpreting population-related data.
For example, a quadratic equation like Japan's population model is differentiated to find a population's growth rate at particular points in time. Similarly, integration could be used to find the total change in population over a specific interval. These calculus tools not only aid in current population studies but are also essential for forecasting future demographic trends, providing an empirical basis for strategic planning in areas like urban development, healthcare, and education.
For example, a quadratic equation like Japan's population model is differentiated to find a population's growth rate at particular points in time. Similarly, integration could be used to find the total change in population over a specific interval. These calculus tools not only aid in current population studies but are also essential for forecasting future demographic trends, providing an empirical basis for strategic planning in areas like urban development, healthcare, and education.
Mathematical Modeling
Mathematical modeling is the process of constructing equations to simulate real-world phenomena. Models allow us to make predictions and to understand more deeply how systems behave.
In our exercise, Japan's population dynamics have been modeled using a quadratic equation. This kind of mathematical model simplifies complex, real-world behavior into an understandable form. By relating the variable \( t \) to year, the model reflects changes in population over time, providing a tool to visualize and calculate various demographic scenarios. It's important, however, to remember that models are simplifications and may not capture every aspect of reality. They need to be periodically revised as new data become available.
In our exercise, Japan's population dynamics have been modeled using a quadratic equation. This kind of mathematical model simplifies complex, real-world behavior into an understandable form. By relating the variable \( t \) to year, the model reflects changes in population over time, providing a tool to visualize and calculate various demographic scenarios. It's important, however, to remember that models are simplifications and may not capture every aspect of reality. They need to be periodically revised as new data become available.
Other exercises in this chapter
Problem 34
Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{3 x^{3}+4 x} $$
View solution Problem 34
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\left(x^{5}-3 x\right)\left(\frac{1}{x^{2}}\ri
View solution Problem 34
Use the limit definition to find the derivative of the function. $$ f(x)=\sqrt{x+2} $$
View solution Problem 34
Find the limit. $$ \lim _{x \rightarrow-1} \frac{4 x-5}{3-x} $$
View solution