Problem 54 - Algebra 1: Concepts and Skills

Question

The United States Bureau of the Census predicts that the population of Florida will be about 17.4 million in 2010 and then will increase by about 0.22 million per year until 2025. Choose the equation that predicts the population y of Florida (in millions) in terms of x, the number of years after 2010. $$ (A) y=17.4 x+0.22 $$ $$ (B)y=-0.22 x+17.4 $$ $$ (C)y=0.22 x+17.4 $$ $$ (D)y=-17.4 x+0.22 $$

Step-by-Step Solution

Answer

The correct equation that predicts the population of Florida in terms of x, the number of years after 2010, is option (C) \(y = 0.22 x + 17.4\). This equation accurately takes into account the linear increase in population over the years since 2010 and the starting population in 2010.

Step 1: Identify the slope and y-intercept

First, we must identify the slope and the y-intercept based on the problem. The slope represents the rate of the population increase per year, which is 0.22 million per year. The y-intercept, on the other hand, is the population of Florida in the year 2010, which is 17.4 million.

Step 2: Formulate the general form of the linear equation

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope of the line, \(b\) is the y-intercept, \(x\) is the independent variable (in this case, the number of years after 2010), and \(y\) is the dependent variable (in this case, the population). Substituting the slope and y-intercept values from Step 1, we get \(y = 0.22x + 17.4\).

Step 3: Match the equation to the provided options

Now, we compare our derived equation to the provided options. Clearly, option (C) \(y = 0.22x + 17.4\) is the equation that correctly predicts the population of Florida in terms of x, the number of years after 2010.

Key Concepts

Slope and Y-Intercept

Understanding the slope and y-intercept is fundamental when dealing with linear equations. Let's dive into these concepts using a real-world example.
In our exercise, the slope, denoted as 'm' in the linear equation format, corresponds to the rate of change in the population of Florida. In other words, it is the amount by which the population increases each year, which is given as 0.22 million per year. The y-intercept, denoted as 'b', represents the starting value of the dependent variable when your independent variable equals zero. In the context of our problem, the y-intercept is the population size at the starting year, which is 2010, and is numerically valued at 17.4 million.

• Slope (m): Indicates the rate of population increase, 0.22 million people per year.
• Y-Intercept (b): Represents the initial population of Florida in 2010, 17.4 million people.

One can visualize this scenario as if you're looking at a graph plotting the population over time. The point where the line representing population growth would intersect the y-axis is where the year is 'zero', or in our case, the year 2010. This intersection point's y-value is the y-intercept.

Linear Equation Formulation

Formulating a linear equation is like building a bridge from a theoretical understanding of slope and y-intercept to their practical application.
To create a linear equation, you will usually start with the standard form, \(y = mx + b\). Here's how you apply it:

1. Identify the slope (rate of change) and the y-intercept (starting value).
2. Insert these values into their respective places in the equation.

For our exercise on population growth, we identified the slope (m) as 0.22 and the y-intercept (b) as 17.4. From there, we plug these values into the equation to get \(y = 0.22x + 17.4\). It's crucial to remember that within this equation, 'x' represents the independent variable (the number of years after 2010), and 'y' is the dependent variable (the population of Florida). This process transforms raw data and information into a predictive tool that can be used for various analytical purposes.

Population Growth Prediction

Predicting population growth using linear equations involves extrapolating future values based on a constant rate of change. It's used by demographers and policy-makers to make strategic decisions based on anticipated demographic trends.
In our case, once we have established our linear equation \(y = 0.22x + 17.4\), we can predict the population for any given year after 2010. For example, to forecast the population for the year 2025, we would calculate the number of years since 2010, which is 15 years. Inserting that 'x' value into our equation, we would get \(y = 0.22\times15 + 17.4 = 20.7\) million people.

Factors Influencing Population Growth

It's important to note that this model assumes a steady, unchanging rate of population growth, which may not always hold true in complex, real-world situations. Factors such as migrations, policy changes, economic shifts, and natural events can affect population numbers in ways that a simple linear equation might not accurately capture. Nonetheless, this approach does provide a useful starting point for estimating future demographic changes.