Question

Use the following information. In \(1991,\) the population of Kenosha, Wisconsin, was \(132,000 .\) Between 1991 and 1996 , the population of Kenosha increased by approximately 2000 people per year. Write an equation that models the population \(y\) of Kenosha in terms of \(x\) where \(x\) represents the number of years since 1991.

Short answer

The equation that models the population \(y\) of Kenosha in terms of \(x\) (where \(x\) represents the number of years since 1991) is \(y=132,000 + 2000x\).

Step-by-step solution

Understanding the Problem

The problem is about a linear increase in population. In 1991, the population was 132,000 and it grew by 2000 people per year. We need to find an equation that describes this situation. \(x\) represents the years since 1991 and \(y\) the population.

Formulating the Equation

In 1991 (when \(x=0\)), the population (\(y\)) was 132,000. Since the population's growing by 2000 people per year, the equation to represent the population growth is \(y=132,000 + 2000x\). This equation expresses the population \(y\) as an initial value of 132,000 with an increase of 2000 for each year \(x\) past 1991.

Key concepts

Linear Equations

When we talk about linear equations, we are referring to a relationship between two variables that when graphed, forms a straight line. This is generally represented in the format of \(y = mx + b\), where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope, and \(b\) is the y-intercept.

In the context of population growth in Kenosha, Wisconsin, the equation \(y = 132,000 + 2000x\) is a practical example of a linear equation. Here, \(y\) represents the population in a given year, and \(x\) denotes the number of years since 1991. The number 132,000 is the y-intercept, which signifies the initial population, and 2000 is the slope, indicating an annual increase in population.

The concept of linear growth here is the constant rate of increase—2000 people per year—which is a key characteristic of linear relationships: the change in y for each unit increase in x is constant.

Algebraic Modeling

Algebraic modeling is a methodology whereby we use algebra to build models that represent real-world situations. In algebraic modeling, we translate given information into algebraic expressions or equations to analyze and solve problems.

In our Kenosha population scenario, we are using algebraic modeling to create an equation that represents the population growth. This model takes the initial population and combines it with the consistent annual increase to give us a formula that can predict future population based on the passing of time since 1991.

By creating and using these models, we can make informed predictions without having to observe the actual growth over time. This is incredibly useful for planners and policymakers who need to account for population changes when making decisions on resources, services, and infrastructure.

Population Mathematics

Population mathematics involves using mathematical concepts and models to understand and predict changes in population numbers. Linear population growth, like that of Kenosha's, is one of the simplest forms of population modeling. It assumes a constant rate of change, which is not influenced by the current population size or other external factors.

In more complex situations, population growth can be modeled using non-linear equations to account for factors such as limited resources, disease, and emigration. However, for Kenosha between 1991 and 1996, the exercise assumes a linear model is sufficient.

When analyzing population trends using linear models, it is crucial to remember that these are simplifications and real-world populations are subject to a wide array of influencing factors. However, such models can still offer valuable insights, especially over short time frames or when the factors affecting growth remain relatively constant.