Problem 38 - Algebra 1

Question

Use the following information. The U.S. Bureau of the Census predicted that the population of Florida would be about 17.4 million in 2010 and then would increase by about 0.22 million per year until 2025 Choose the linear model that predicts the population \(P\) of Florida (in millions) in terms of \(t,\) the number of years since 2010 . A) \(P=17.4 t+0.22\) (B) \(P=-0.22 t+17.4\) (c) \(P=0.22 t+17.4\) (D) \(P=-17.4 t+0.22\)

Step-by-Step Solution

Answer

The correct linear model to predict the population of Florida, \(P\), in terms of \(t,\) is (C) \(P=0.22 t+17.4\).

Step 1: Understand the initial value

For a linear function of the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept, the y-intercept is the initial value, i.e., the initial population of Florida in 2010, so it is found in the equation where the constant term is 17.4.

Step 2: Understand the rate of increase

In the same function, the slope \(m\) is the rate of change. Here, the rate of population increase per year is the coefficient of \(t\). This means it is found in the equation where the coefficient of \(t\) is 0.22.

Step 3: Select the Model

Applying this to the options given, choice A does not reflect a rate of change per year, as it would mean the population is multiplied by the number of years, which is not the situation. Legacy choice B gives a decreasing population, not an increasing one. Choice C presents an increasing population and starts at 17.4 million, so it is a good fit for the given situation. Finally, choice D again does not reflect the situation and also gives a decreasing population. Thus, choice C: \(P=0.22t+17.4\) is the correct model.

Key Concepts

Population Growth

Population growth is the change in the number of people living in a particular area over time. In this context, we are discussing the population increase in Florida from the year 2010 onwards. The initial population is 17.4 million, and there is an expected steady growth.
The provided linear model helps to predict this growth over the years by using a constant rate of increase each year. When we talk about population growth in a mathematical sense, we simplify reality by assuming that growth occurs in a straight line, hence the term "linear models."
To visualize this:

• In 2010, the population is 17.4 million.
• In 2011, it becomes 17.4 + 0.22 million.
• In 2012, it becomes 17.4 + 0.22 * 2 million, and so on.

This model assumes an ongoing rate without fluctuation, which is a straightforward way to estimate future population when detailed data is not available.

Slope and Y-intercept

The linear equation takes the form of \[ y = mx + b \] where:

• \( m \) represents the slope, indicating how much the population changes per year.
• \( b \) is the y-intercept, representing the starting value or initial population at the base year of 2010.

The slope is crucial because it tells us how steeply the population is increasing. In this case, it's an increase of 0.22 million people each year. This linear representation is helpful because it simplifies complex real-world growth into something we can work with algebraically.
The y-intercept, on the other hand, helps us determine the starting point on a graph. For Florida's population scenario, it is the figure of 17.4 million, set in the year 2010. This makes our entire equation and graph applicable from this specific point in the past.

Algebraic Equations

Algebraic equations, like the one used to predict Florida's population growth, are fundamental tools in mathematics to express relationships. With the equation \[ P = 0.22t + 17.4 \] we are defining a relationship between two variables:

• \( P \), the population in millions.
• \( t \), the time in years since 2010.

This equation uses variables and constants to express how population changes over time. Algebra makes it possible to find the population for any year after 2010 by plugging in a particular year to the equation as \( t \).
If you want to know the population in 2020, for example, you substitute \( t = 10 \) into our equation, allowing us to calculate the population by simply performing arithmetic operations. It's a versatile tool for making projections based on given data.