Problem 37 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

DEMOGRAPHICS Demographic studies indicate that the fraction of the residents that will remain in a certain city for at least \(t\) years is \(f(t)=e^{-t / 20}\). The current population of the city is 100,000 , and it is estimated that \(t\) years from now, new people will be arriving at the rate of \(100 t\) people per year. If this estimate is correct, what will happen to the population of the city in the long run?

Step-by-Step Solution

Answer

The population will grow quadratically over time.

Step 1: Understand the Retention Function

The fraction of residents that will remain in the city for at least \(t\) years is given by \(f(t) = e^{-t / 20}\). This function shows an exponential decay in the number of residents over time.

Step 2: Define Current Population

The current population of the city is 100,000. This will act as our starting point for calculating changes in population over time.

Step 3: Define the Influx of New Residents

New people are arriving at the rate of \(100t\) people per year. This means after \(t\) years, the number of new arrivals is \int_{0}^{t} 100t \, dt = 50t^2\.

Step 4: Calculate Retained Residents

The population retained from the initial residents after \(t\) years is \100{,}000 \times e^{-t / 20}\.

Step 5: Calculate Total Population Over Time

Sum the retained residents and the new arrivals to get the total population \(P(t) = 100{,}000e^{-t/20} + 50t^2\).

Step 6: Determine Long-Term Behavior

As \(t \rightarrow \infty\), the term \100{,}000e^{-t/20} \rightarrow 0\ and the term \50t^2\ dominates. Therefore, the long-term population behavior is approximated by \(P(t) \rightarrow 50t^2\).

Key Concepts

Exponential Decay

Exponential decay is a crucial concept in understanding population dynamics. In demography, it refers to how a population decreases over time. The function given in the problem, \(f(t) = e^{-t / 20}\), represents the fraction of residents who stay in a city for at least \(t\) years. This type of function shows that the longer the duration, the smaller the fraction of people who remain.

The formula \(e^{-t/20}\) indicates that the population decreases rapidly at first, then the rate of decay slows down over time. This exponential decay is characterized by a constant percentage rate of decrease over equal time intervals.

To understand this better, consider a hypothetical starting with 100% of the population. After 20 years, only about 36.8% remain, and after 40 years, only about 13.5% are still in the city. The use of \(e^{-t/20}\) provides a clear mathematical model that can be applied to various demographic studies for predicting population retention.

Population Retention

Population retention is the measure of how many people stay in a location over a certain period of time. The initial population is often subject to various factors like migration, birth, and death rates, which can influence the retention rate.

In the given problem, the retention function is modeled by \(f(t) = e^{-t/20}\). If the initial population is 100,000, the number of residents remaining after \(t\) years is given by \(100,000 \times e^{-t/20}\).

This means that as time progresses, fewer and fewer people from the original population stay in the city. This model helps us understand long-term population trends and aids in city planning and resource allocation.

For example, after 20 years, approximately 36,788 people would remain from the initial population of 100,000. This kind of retention analysis is fundamental for making informed decisions in public policy and infrastructure development.

Integration of Arrival Rate

When considering the dynamics of a population, it's also essential to account for new arrivals. The arrival rate function given in the problem is \(100t\), indicating that the number of new residents arriving at time \(t\) grows linearly with time.

To find the total number of new arrivals over a period, we need to integrate this rate from 0 to \(t\). The integral of the arrival rate, \(\[\begin{equation} \int_{0}^{t} 100t \ dt \end{equation}\]\), evaluates to \(50t^2\). This result tells us how many new people have moved to the city by time \(t\).

Combining this with the retention function allows us to calculate the total population at any given time. The formula becomes:
\( P(t) = 100,000 e^{-t/20} + 50t^2 \)

This equation shows that the retained part of the population declines over time, while the new arrivals increase quadratically. Long-term, the quadratic term dominates, implying a growing population as time advances. Understanding these integrations is key in demography for predicting future population sizes and trends.