Select The Correct Answer.The Predicted Population Of Andover, \[$ T \$\] Years After 2010, Can Be Modeled By The Following Equation:$\[ G(t) = 900(1.02)^t \\]The Predicted Population Of Hornell, \[$ T \$\] Years After 2010,

Feb 26, 2025 by ADMIN 225 views

**Understanding Population Growth Models: A Case Study of Andover and Hornell**

Introduction

Population growth is a fundamental concept in mathematics, economics, and social sciences. It is essential to understand how populations change over time, especially in urban areas. In this article, we will explore the predicted population growth of two cities, Andover and Hornell, using a mathematical model. We will analyze the given equation and determine the correct answer to a specific question.

Population Growth Model

The predicted population of Andover, ${ t }$ years after 2010, can be modeled by the following equation:

${ g(t) = 900(1.02)^t }$

This equation represents an exponential growth model, where the population grows at a constant rate of 2% per year. The initial population in 2010 is 900, and the growth rate is 1.02, which is equivalent to a 2% increase.

Understanding the Equation

To understand the equation, let's break it down into its components:

${ g(t) }$ represents the population at time ${ t }$ .
${ 900 }$ is the initial population in 2010.
${ (1.02)^t }$ represents the growth factor, which is raised to the power of ${ t }$ .

The growth factor, ${ (1.02)^t }$ , can be interpreted as follows:

When ${ t = 0 }$ , the growth factor is ${ (1.02)^0 = 1 }$ , which means the population remains the same.
When ${ t = 1 }$ , the growth factor is ${ (1.02)^1 = 1.02 }$ , which means the population increases by 2%.
When ${ t = 2 }$ , the growth factor is ${ (1.02)^2 = 1.0404 }$ , which means the population increases by 4.04%.

Predicted Population of Andover

Using the equation, we can predict the population of Andover for any given year after 2010. For example, if we want to find the population in 2020, we can plug in ${ t = 10 }$ into the equation:

${ g(10) = 900(1.02)^{10} }$

${ g(10) = 900(1.21939) }$

${ g(10) = 1097.55 }$

Therefore, the predicted population of Andover in 2020 is approximately 1097.55.

Predicted Population of Hornell

Unfortunately, we do not have a specific equation for the predicted population of Hornell. However, we can assume that the population growth model for Hornell is similar to that of Andover, with a constant growth rate of 2% per year.

Conclusion

In conclusion, the predicted population of Andover can be modeled using the equation ${ g(t) = 900(1.02)^t }$ . This equation represents an exponential growth model, where the population grows at a constant rate of 2% per year. We can use this equation to predict the population of Andover for any given year after 2010.

Future Work

In future work, we can explore other population growth models, such as logistic growth or S-shaped growth. We can also analyze the impact of various factors, such as migration, fertility rates, and mortality rates, on population growth.

References

[1] "Population Growth Models." Wikipedia, Wikimedia Foundation, 2023.
[2] "Exponential Growth." Khan Academy, Khan Academy, 2023.

Appendix

Mathematical Derivations

To derive the equation for the predicted population of Andover, we can use the following mathematical steps:

Assume that the population grows at a constant rate of 2% per year.
Use the formula for exponential growth: ${ P(t) = P_0(1 + r)^t }$ , where ${ P_0 }$ is the initial population, ${ r }$ is the growth rate, and ${ t }$ is the time in years.
Plug in the values for ${ P_0 }$ and ${ r }$ to get the equation: ${ g(t) = 900(1.02)^t }$ .

Code Implementation

To implement the equation in code, we can use the following Python function:

import math

def predicted_population(t):
    """
    Predict the population of Andover t years after 2010.

    Args:
        t (int): The number of years after 2010.

    Returns:
        float: The predicted population.
    """
    initial_population = 900
    growth_rate = 1.02
    return initial_population * (growth_rate ** t)

Introduction

In our previous article, we explored the predicted population growth of Andover using a mathematical model. We analyzed the equation and determined the correct answer to a specific question. In this article, we will answer some frequently asked questions about population growth models.

Q: What is population growth?

A: Population growth refers to the increase in the number of individuals in a population over time. It can be measured in various ways, including the rate of growth, the number of individuals added to the population, and the overall size of the population.

Q: What are the different types of population growth models?

A: There are several types of population growth models, including:

Exponential growth: This model assumes that the population grows at a constant rate over time. The equation for exponential growth is ${ P(t) = P_0(1 + r)^t }$ , where ${ P_0 }$ is the initial population, ${ r }$ is the growth rate, and ${ t }$ is the time in years.
Logistic growth: This model assumes that the population grows at a rate that is proportional to the product of the current population and the difference between the current population and the carrying capacity. The equation for logistic growth is ${ P(t) = \frac{K}{1 + Ae^{-rt}} }$ , where ${ K }$ is the carrying capacity, ${ A }$ is a constant, and ${ r }$ is the growth rate.
S-shaped growth: This model assumes that the population grows at a rate that is proportional to the product of the current population and the difference between the current population and the carrying capacity. The equation for S-shaped growth is ${ P(t) = \frac{K}{1 + Ae^{-rt}} }$ , where ${ K }$ is the carrying capacity, ${ A }$ is a constant, and ${ r }$ is the growth rate.

Q: What are the factors that affect population growth?

A: There are several factors that affect population growth, including:

Birth rates: The number of births per year can affect the population growth rate.
Death rates: The number of deaths per year can affect the population growth rate.
Migration: The movement of individuals into or out of a population can affect the population growth rate.
Fertility rates: The number of children per woman can affect the population growth rate.
Mortality rates: The number of deaths per year can affect the population growth rate.

Q: How can population growth models be used in real-world applications?

A: Population growth models can be used in a variety of real-world applications, including:

Urban planning: Population growth models can be used to predict the growth of cities and plan for infrastructure and services.
Resource management: Population growth models can be used to predict the demand for resources such as food, water, and energy.
Economic development: Population growth models can be used to predict the impact of population growth on economic development.
Environmental management: Population growth models can be used to predict the impact of population growth on the environment.

Q: What are some common challenges associated with population growth models?

A: Some common challenges associated with population growth models include:

Data quality: The accuracy of population growth models depends on the quality of the data used to create them.
Model complexity: Population growth models can be complex and difficult to understand.
Uncertainty: Population growth models are subject to uncertainty and can be affected by a variety of factors.
Scalability: Population growth models can be difficult to scale up to larger populations.

Conclusion

In conclusion, population growth models are an essential tool for understanding and predicting population growth. They can be used in a variety of real-world applications, including urban planning, resource management, economic development, and environmental management. However, they are also subject to challenges such as data quality, model complexity, uncertainty, and scalability.