Population Density The population density of the United...

Population Density The population density of the United States has been steadily increasing since 2000 . The population density of the United States can be estimated using the equation $$ P=0.82 t+78.5 $$ where $P$ is the population density, measured in people per square mile, and $t$ is the number of years since 2000. Use $t=1$ for 2001, $t=2$ for 2002, and so on. If the population density continues to increase at its current rate, a) determine the population density of the United States in $2018 .$ b) during what year will the population density of the United States reach 100 people per square mile?

Population Density The population density of the United States has been steadily increasing since 2000 . The population density of the United States can be estimated using the equation
$$
P=0.82 t+78.5
$$
where $P$ is the population density, measured in people per square mile, and $t$ is the number of years since 2000. Use $t=1$ for 2001, $t=2$ for 2002, and so on. If the population density continues to increase at its current rate,
a) determine the population density of the United States in $2018 .$
b) during what year will the population density of the United States reach 100 people per square mile?

Key Concepts

Linear Functions

Linear functions are mathematical models that express a relationship between two variables with a constant rate of change. They are typically written in the form y = mx + b, where m represents the slope and b represents the y-intercept. In real-world problems, linear functions are used to predict future values based on known trends, making them useful for estimating quantities like population density over time.

Rate of Change (Slope)

The slope in a linear function indicates how much the dependent variable changes for a one?unit change in the independent variable. In practical contexts, it represents a constant rate of change over time, such as the annual increase in population density. Understanding the slope allows one to quantify the increase or decrease of a quantity as time progresses.

Y-Intercept

The y-intercept is the constant term in the linear function where the graph crosses the y-axis. It represents the initial value of the quantity when the independent variable is zero. In modeling scenarios, this initial condition helps establish a baseline from which future values are calculated.

Evaluation of Functions

Evaluating a function involves substituting a specific value for the independent variable to compute the corresponding value of the dependent variable. This is crucial in applying mathematical models to determine outcomes at specific points in time, such as finding the estimated value of a population in a given year.

Solving Linear Equations

Solving a linear equation involves isolating the variable of interest to find its value when the function reaches a particular outcome. This technique is essential in applications where one needs to predict when a certain level or threshold will be achieved, such as determining the year when a population density will meet or exceed a specified value.

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