Problem 26 - Precalculus

Question

During the early years of a community, it is not uncommon for the population to grow according to the Law of Uninhibited Growth. According to the Painesville Wikipedia entry, in \(1860,\) the Village of Painesville had a population of \(2649 .\) In \(1920,\) the population was 7272\. Use these two data points to fit a model of the form \(N(t)=N_{0} e^{k t}\) were \(N(t)\) is the number of Painesville Residents \(t\) years after \(1860 .\) (Use \(t=0\) to represent the year 1860 . Also, round the value of \(k\) to four decimal places.) According to this model, what was the population of Painesville in \(2010 ?\) (The 2010 census gave the population as 19,563 ) What could be some causes for such a vast discrepancy? For more on this, see Exercise 37 .

Step-by-Step Solution

Answer

To find the population in 2010, use the model \( N(150) = 2649 \times e^{0.0167 \times 150} \). Changes in growth rates could explain discrepancies with the 2010 census of 19,563.

Step 1: Identify Known Quantities

We are given two data points: in 1860, the population \( N_0 = 2649 \), and in 1920, the population \( N(60) = 7272 \) since 1920 is 60 years after 1860.

Step 2: Write the Initial Model Equation

The model for population growth is \( N(t) = N_0 e^{kt} \). Here, \( N_0 = 2649 \). We need to find the value of \( k \).

Step 3: Solve for the Growth Rate Constant "k"

Substitute the known values into the equation for the year 1920: \( 7272 = 2649 e^{60k} \). Rearrange to solve for \( k \):\[ e^{60k} = \frac{7272}{2649} \]\[ 60k = \ln \left( \frac{7272}{2649} \right) \]\[ k = \frac{\ln \left( \frac{7272}{2649} \right)}{60} \].Calculate \( k \) using a calculator, round to four decimal places: \( k \approx 0.0167 \).

Step 4: Model the Population for 2010

2010 is 150 years after 1860, so \( t = 150 \). Use the model \( N(t) = N_0 e^{kt} \) with \( k = 0.0167 \), \( N_0 = 2649 \):\[ N(150) = 2649 \times e^{0.0167 \times 150} \].Calculate \( N(150) \) to estimate 2010 population.

Step 5: Evaluate Discrepancy Causes

Compute \( N(150) \) and compare it to the 2010 census value (19,563). Discrepancies may arise from changes in growth rates over time, economic impacts, migration, or changes in birth/death rates.

Key Concepts

Law of Uninhibited Growth

The Law of Uninhibited Growth describes how a population expands when it faces no constraints. It assumes unlimited resources, no predators, and ideal growth conditions. In real-world scenarios, this depicts an early growth phase of a population, such as a newly established settlement or colony.

Mathematically, the law is modeled by the equation:

• \( N(t) = N_0 e^{kt} \)

where:

• \( N(t) \) is the population at time \( t \).
• \( N_0 \) is the initial population size.
• \( k \) is the growth rate constant.
• \( t \) is the time elapsed since the starting point.

This model helps in understanding theoretical growth potential without limits, emphasizing the importance of \( k \), which governs growth speed.

Population Modeling

Population modeling is a critical method in ecology and urban planning, used to estimate how a population changes over time. By using mathematical equations like \( N(t) = N_0 e^{kt} \), we can predict future growth trends. This is particularly useful for planning resources, such as schools or hospitals.

In practice, population modeling provides:

• Insight into short-term and long-term population changes.
  
  
• Tools to test different hypothetical scenarios on population effects, such as policy changes.
  
  
• A framework to identify key variables affecting population dynamics, including birth/death rates and migration.

Accurate models rely on historical data and can be adjusted as new information becomes available, ensuring they remain relevant and practical.

Growth Rate Constant

The growth rate constant, denoted as \( k \), is a crucial factor in exponential models. This constant determines how rapidly the population grows or shrinks over time. The larger the \( k \), the faster the growth.

To calculate \( k \), you use the formula:

• \( k = \frac{\ln \left( \frac{N(t)}{N_0} \right)}{t} \)

This formula relates the initial population \( N_0 \), the population at a later time \( N(t) \), and the time \( t \).

In our example, the calculation revealed a \( k \approx 0.0167 \), indicating a modest growth between 1860 and 1920. Understanding \( k \) allows planners and ecologists to make informed predictions about future population sizes.

Population Discrepancy Causes

Population discrepancies arise when there is a significant difference between modeled predictions and real-world data. In the Painesville exercise, the model estimation for 2010 differed from the census data. These variances usually occur due to several factors:

• Changes in growth rates over time. The model assumes a constant \( k \), but in reality, growth rates can fluctuate due to economic or social shifts.
  
  
• Migration. Unpredictable patterns of people moving in and out can dramatically affect local populations.
  
  
• Environmental changes impacting resources; as conditions become less ideal, growth may slow.
  
  
• Policy changes that affect birth rates, immigration, or emigration.

Thus, models need continuous updating to account for these factors and even then, remain approximations rather than precise predictions.