Question

S POPULATION In 1990 Tampa, Florida, had a population of \(280,015\). From 1990 through \(1996,\) the population increased at an average rate of about 850 per year. Write a linear model for the population of Tampa. Source: U.S. Bureau of the Census

Short answer

The linear model representing Tampa's population growth from 1990 through 1996 is \(P(t) = 850t + 280,015\), where 'P' is the population size and 't' is the number of years after 1990.

Step-by-step solution

Identifying Given Values

Decode the parameters given in the problem. The initial population in 1990 is 280,015 people. The population growth rate is an increase of approximately 850 people per year.

Understand Linear Concept

Recognize that the population growth is linear. This means that it follows the pattern 'y = mx + c', where 'm' represents the slope (population growth per year) and 'c' represents the y-intercept (the initial population).

Form the Linear Model

Formulate the linear model using the given parameters. Let 'P' represent the population count, 't' represent the time in years (where t = 0 corresponds to 1990), then the model becomes: \(P(t) = 850t + 280,015\)

Key concepts

Understanding Population Growth

When we talk about population growth, we're referring to the change in the number of individuals in a population over time. In many cases, the growth rate of a population can be described using linear models, particularly when the growth is consistent. In such scenarios, the population increases or decreases by a fixed number of individuals each year.

To comprehend population growth in a linear context, we distinguish two key components: the initial population at the start of observation, and the constant rate of growth (or decline). For instance, if a town’s population grows by 500 individuals annually, that’s a linear growth rate. This can be visualized on a graph as a straight line, which helps to predict future population sizes, assuming the growth rate remains constant.

Understanding this concept is crucial in various fields such as ecology, sociology, and urban planning. It assists in revealing trends and preparing for future developments regarding resource allocation, infrastructure projects, and environmental impact assessments.

Linear Equations: The Foundation of Algebraic Predictions

Linear equations form the bedrock of algebraic predictions and are ubiquitously applied in diverse fields of study. The standard form of a linear equation is expressed as \(y = mx + c\), which is an equation of the first degree. Here, \(y\) represents the dependent variable we wish to predict or understand, \(m\) indicates the slope or the rate of change, \(x\) is the independent variable, and \(c\) denotes the y-intercept, the value of \(y\) when \(x\) is zero.

These equations graph as straight lines in the coordinate plane, which is why they're referred to as 'linear'. The slope indicates the direction of the line and the steepness of the climb or descent. In our example, the increase of approximately 850 people per year is the slope of the linear equation that predicts Tampa's population. This mathematical tool enables us to forecast outcomes and make prudent decisions based on the behavior of various variables.

Algebraic Modeling in Real-World Applications

Algebraic modeling is a method of using mathematical equations to represent real-world situations. It involves converting a given scenario into an algebraic expression or equation that can be analyzed or solved mathematically. Through algebraic models, one can simulate situations, predict future outcomes, or analyze trends.

Models can be as simple or complex as necessary depending on the factors at play. In the context of population growth, a linear model provides a straightforward approach to predict future population sizes. For example, the linear equation \(P(t) = 850t + 280,015\) models Tampa's population, illustrating how it might evolve over time if the current rate of growth remains consistent.

By plugging different values for \(t\), representing years since 1990, into the model, you can estimate the population for any subsequent year. Algebraic modeling is not just a theoretical exercise; it is essential for policymaking, budgeting, and strategic planning, especially in sectors that depend on demographic statistics.