Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Global casino gaming revenue from online betting was \(\$ 147\) billion in 2012 and \(\$ 183\) billion in 2015 . (Source: Statista.) (a) Find an equation of a line \(y=m x+b\) that models this information, where \(y\) is in billions of dollars and \(x\) is the year. (b) Use this equation to estimate online betting revenue in 2014

Step-by-Step Solution

Answer

Revenue for 2014 is estimated at \(\$171\) billion.

Step 1: Identify the Points

We have two points to work with: \((2012, 147)\) and \((2015, 183)\). Here, the first coordinate represents the year \(x\) and the second coordinate represents the revenue \(y\) in billions.

Step 2: Calculate the Slope

The slope \(m\) of a line is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using our points \((2015, 183)\) and \((2012, 147)\), the slope is computed as \[m = \frac{183 - 147}{2015 - 2012} = \frac{36}{3} = 12\].

Step 3: Use the Point-Slope Form

With the slope and one of the points, use the point-slope formula \(y - y_1 = m(x - x_1)\). Using the point \((2012, 147)\), we have \[y - 147 = 12(x - 2012)\].

Step 4: Convert to Slope-Intercept Form

Rearrange the equation to get \(y = mx + b\). Starting from \(y - 147 = 12(x - 2012)\), expand and simplify: \[y - 147 = 12x - 24144\] \[y = 12x - 24144 + 147\] \[y = 12x - 23997\].

Step 5: Estimate Revenue for 2014

Substitute \(x = 2014\) into the equation \(y = 12x - 23997\): \[y = 12 \times 2014 - 23997\] \[y = 24168 - 23997 \] \[y = 171\].

Step 6: Review the Result

The estimated online betting revenue for 2014 is \(\$171\) billion, based on the linear model.

Key Concepts

Slope Calculation

To understand how relationships between two variables can be modeled linearly, we begin by computing the slope. The slope is essentially a measure of how one variable changes with respect to another. In the context of linear regression, the slope signifies the rate of change.To calculate the slope, use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \( y_2 \) and \( y_1 \) represent the changes in the dependent variable (revenue, in our case), and \( x_2 \) and \( x_1 \) are the changes in the independent variable (years). For our problem, using \( (2012, 147) \) and \( (2015, 183) \):

• \( y_2 = 183 \) billion, and \( y_1 = 147 \) billion,
• \( x_2 = 2015 \), \( x_1 = 2012 \).
• Calculate \( m \) as \( \frac{183 - 147}{2015 - 2012} = 12 \).

The slope of \( 12 \) indicates the revenue increases by \( 12 \) billion for each year.

Point-Slope Form

Once the slope is known, we can use the point-slope form of a line. This form is useful in creating an equation of a line when a point on the line and its slope are known.The formula for point-slope form is:\[ y - y_1 = m(x - x_1) \]Using the point \( (2012, 147) \) and the slope \( 12 \), we plug the values into the formula:\[ y - 147 = 12(x - 2012) \]This equation represents the line using the specific point and slope. The point-slope form is especially handy for quickly identifying the relationship between variables as it clearly links the slope with a particular point. This is an intermediary step before converting into other forms, like slope-intercept, for simplified expressions.

Slope-Intercept Form

After obtaining the line's equation in point-slope form, we convert it to slope-intercept form for ease of reading and application.The slope-intercept form is:\[ y = mx + b \]This shows the dependent variable \( y \) directly as a function of the independent variable \( x \), slope \( m \), and y-intercept \( b \).Starting with the equation \( y - 147 = 12(x - 2012) \), we transform it:

• Firstly, expand: \( y - 147 = 12x - 24144 \).
• Add 147 to both sides: \( y = 12x - 24144 + 147 \).
• Simplify to: \( y = 12x - 23997 \).

In this form, the slope \( 12 \) and y-intercept \( -23997 \) provide a clear linear function model of the revenue over time. It can be quickly used to predict or estimate values for different \( x \) years.

Estimating Values

The benefit of using a linear equation is its ability to estimate values efficiently, often for unseen data points. Having an equation like \( y = 12x - 23997 \) allows us to quickly compute predictions.To estimate the revenue for 2014:

• Insert \( x = 2014 \) into the equation: \( y = 12 imes 2014 - 23997 \).
• Calculate: \( y = 24168 - 23997 \).
• Simplify to find \( y = 171 \).

Thus, our model estimates the 2014 online betting revenue to be \( 171 \) billion dollars. Estimating future or unknown values involves substituting the relevant \( x \) values into your equation, allowing for quick predictions.

Linear Modeling

Linear modeling involves building a mathematical model that creates a straight-line representation of the relationships between variables. The advantage of linear modeling is in its simplicity and its ability to provide insights with minimal data.In our scenario, we're modeling revenue against the year using a linear relationship. Our model, \( y = 12x - 23997 \), derived from two data points, doesn't just reflect those values but acts as a prediction tool.This approach is:

• Simple to compute and understand, making it perfect for initial analysis.
• Useful for quick predictions over short periods assuming linear growth.
• Based on assumptions that relationships are steady and linear, which may not always hold over long durations or with complex systems.

Linear models are powerful for making estimates and forming an initial understanding of how variables might interact over a series of events or periods in time.