In 2004, the magazine had a circulation of approximately 1 million readers, which increased to about 2 million by 2014. To create a linear model representing the magazine's readership, \( r \), \( t \) years after 2000, follow these steps: 1. Calculate the annual rate of change in the number of readers (the slope). 2. Consider the year 2000 as the starting point (\( t = 0 \)) and use the provided readership data for subsequent years to find the slope. Between 2004 and 2014, a period of 10 years, readership rose by 1 million. Calculate the slope (\( m \)) by dividing the increase in readership by the time elapsed: \( m = \frac{\text{change in readership}}{\text{change in time}} = \frac{2,000,000 - 1,000,000}{2014 - 2004} = \frac{1,000,000}{10} = 100,000 \) This slope signifies a yearly increase of 100,000 readers. 3. Find the \( y \)-intercept (\( b \)), which is the 2000 readership number (\( t = 0 \)). In 2004 (\( t = 4 \)), there were 1 million readers. Using the slope and the year, determine \( b \): \( 1,000,000 = 100,000 \times 4 + b \) \( 1,000,000 = 400,000 + b \) \( b = 1,000,000 - 400,000 \) \( b = 600,000 \) With the slope and \( y \)-intercept known, the linear model is: \( r = 100,000t + 600,000 \) Here, \( r \) is the number of readers, and \( t \) the years since 2000.

To create the linear model for the magazine readership, follow these steps:

1. Calculate the slope (annual rate of change in readership): This represents the change in the number of readers per year. To calculate it, you use the information about how much the readership changed over a specific time period.

Using the data provided: - In 2004, there were 1 million readers. - In 2014, there were 2 million readers.

The change in readership over this 10-year period is 2 million - 1 million = 1 million readers.

Since 2014 is 14 years after 2000 and 2004 is 4 years after 2000, the time elapsed (change in time) between the two data points is 14 - 4 = 10 years.

Using the formula for the slope (m): \( m = \frac{\text{change in readership}}{\text{change in time}} \) \( m = \frac{1,000,000 \text{ readers}}{10 \text{ years}} \) \( m = 100,000 \text{ readers per year} \)

2. Find the y-intercept (b), the readership number in the base year (2000, when t = 0). To find the y-intercept using the year 2004 as a reference point, plug the values for that year into the linear equation \( r = mt + b \):

\( r \) is the number of readers in 2004, which is 1 million. \( t \) is 4 since 2004 is 4 years after 2000. \( m \) is the slope, which we calculated as 100,000.

So the equation with the known values becomes: \( 1,000,000 = 100,000(4) + b \)

Now solve for \( b \): \( 1,000,000 = 400,000 + b \) \( b = 1,000,000 - 400,000 \) \( b = 600,000 \)

The y-intercept, or the number of readers in the year 2000, is 600,000.

3. With the slope and y-intercept known, the linear model representing the magazine's readership, \( r \), in terms of \( t \), years after 2000, is: \( r = 100,000t + 600,000 \)

This model can be used to estimate the readership for any given year after 2000 by plugging in the value of \( t \) into the equation.

Extra: In general, a linear model is an algebraic equation that describes a relationship between two quantities that vary directly with one another. In this case, the relationship is between the year (after 2000) and the magazine’s readership.

In the linear model equation \( r = mt + b \):

- \( r \) is the dependent variable (readership in this case) because it depends on the independent variable \( t \) (time in years since 2000). - \( t \) represents time in years since the year 2000. - \( m \) is the slope of the line, representing the rate of change in readership per year. It determines how steep or flat the line in a graph will be. - \( b \) is the y-intercept, representing the readership at the starting point (year 2000). It's where the line will cross the y-axis on a graph.

The slope of the line (m) indicates the trend of readership over time. If \( m \) is positive, as it is in this case, it means that the readership is increasing each year. On the other hand, if \( m \) were negative, it would indicate a decrease in readership yearly.

Linear models are helpful in making predictions, understanding trends, and analyzing patterns over time. This model assumes that the rate of change in readership remains constant over time, which won't always be the case in real-world scenarios—there may be years with significant fluctuations that wouldn't perfectly fit a straight line. It's always essential to recognize the limitations of any model and understand it's a simplification that helps us grasp the underlying trends of complex phenomena.