The population of the United States since 2010 can be...

The population of the United States since 2010 can be estimated by the equation $P=2.76 t+309.37,$ where $P$ is the population in millions $t$ years since $2010 .$ Source: Based on data from the U.S. Census Bureau. a. What was the population of the United States in $2013 ?$ b. In what year, was the population of the United States $326,000,000 ?$ c. In what year will the population of the United States reach 375 million?

The population of the United States since 2010 can be estimated by the equation $P=2.76 t+309.37,$ where $P$ is the population in millions $t$ years since $2010 .$ Source: Based on data from the U.S. Census Bureau.
a. What was the population of the United States in $2013 ?$
b. In what year, was the population of the United States $326,000,000 ?$
c. In what year will the population of the United States reach 375 million?

Key Concepts

Linear Functions

A linear function describes a relationship between two variables with a constant rate of change and is typically expressed in the form f(x) = mx + b. This form indicates that for each unit increase in the independent variable, the dependent variable changes by a constant multiple, represented by the slope (m), plus an initial value given by the y-intercept (b).

Slope and Y-Intercept

In the equation of a linear function, the slope is the coefficient that indicates the rate at which the dependent variable changes as the independent variable increases. The y-intercept is the value of the function when the independent variable is zero, representing the starting point of the model. These two components allow for both the prediction of values and the interpretation of relationships between the variables.

Solving Linear Equations

Solving a linear equation typically involves isolating the variable of interest by performing inverse operations such as addition, subtraction, multiplication, or division. This process is fundamental in determining unknown values when some components of a linear model are given.

Modeling Real-World Phenomena

Real-world modeling uses mathematical functions to predict or describe behaviors observed in natural or social phenomena. Linear models, in particular, are used to represent situations where changes occur uniformly, making them a useful tool for estimating future values or understanding trends over time.

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