Problem 53 - College Algebra with Modeling and Visualization

Question

Exercises \(49-54:\) Write a formula for a linear function that models the situation. Choose both an appropriate name and an appropriate variable for the function. State what the input variable represents and the domain of the function. Assume that the domain is an interval of the real numbers. Population Density In 1900 the average number of people per square mile in the United States was \(21.5,\) and it increased, on average, by 5.81 people every 10 years until 2000 . (Source: Bureau of the Census.)

Step-by-Step Solution

Answer

The linear function is \( P(t) = 0.581t + 21.5 \) with domain \( [0, 100] \).

Step 1: Understand the Given Information

The problem provides two key pieces of information: the initial population density in 1900 was 21.5 people per square mile, and the average increase in population density was 5.81 people every 10 years.

Step 2: Define the Variables

Let's define the input variable. Let "\( t \)" represent the number of years since 1900. Thus, \( t = 0 \) corresponds to the year 1900. The output, population density, can be modeled as a function of \( t \). We'll name this function \( P(t) \).

Step 3: Write the Equation for the Linear Function

The linear function can be written as \( P(t) = mt + b \), where \( m \) is the slope representing the rate of increase in population density, and \( b \) is the y-intercept representing the initial population density. From the information given, \( m = \frac{5.81}{10} = 0.581 \) and \( b = 21.5 \). Therefore, the equation is \( P(t) = 0.581t + 21.5 \).

Step 4: Determine the Domain of the Function

The function models the situation from 1900 to 2000. We denote the domain using the number of years since 1900, so \( t \) ranges from 0 to 100, inclusive. Thus, the domain is \( [0, 100] \).

Step 5: State the Input Variable

The input variable \( t \) represents the number of years since 1900, capturing time progression over the century.

Key Concepts

Population Density

Population density is an important concept in geography and demography. It refers to the number of people living per unit of area, typically per square mile or square kilometer. Understanding population density helps in analyzing how crowded a place is and in making decisions related to urban planning and resource distribution.
The original exercise involves the population density of the United States, which was 21.5 people per square mile in 1900. Over the years, various factors such as birth rates, immigration, and migration patterns influenced how quickly or slowly this density increased. Between 1900 and 2000, it rose on average by 5.81 people every 10 years.
This linear increase allows us to model population density using a linear function, capturing how the density changes over time. By examining and solving such exercises, we can get a clearer understanding of how historical trends continue to shape current population distributions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the backbone of many mathematical models, including linear functions. In our exercise, we defined the linear function representing population density as an algebraic expression.
A linear function can be represented as \(P(t) = mt + b\). Here, \(mt\) refers to the term that changes with time, and \(b\) is a constant term representing the starting point.

• The slope \(m\) is \(0.581\), showing how much the population density changes every year (since it's derived from an increase of 5.81 people every 10 years).
• The y-intercept \(b\) is \(21.5\), representing the initial population density in 1900.

In breaking down the algebraic expression for this problem, students see how changes over time can be systematically represented and analyzed using algebra.

Domain of the Function

The domain of a function refers to all the possible input values (the x-values) that a function can accept. In our linear function model for population density, we must determine the domain based on the meaningful time frame of the data provided.
For this exercise, the function is meant to describe changes between 1900 and 2000. Therefore, the input variable \(t\), representing the years since 1900, should only take values within these years. This gives us a domain of \( [0, 100] \), inclusive of the years 1900 (t = 0) to 2000 (t = 100).
Having this clearly defined domain ensures that the model's predictions remain relevant and accurate, strictly covering the timeline it was designed for without extrapolating beyond it. Understanding domain is crucial in determining where a model's application is valid.