Problem 90 - College Algebra with Modeling and Visualization

Question

Online Betting Consumer gambling losses from online betting were \(\$ 4\) billion in 2002 and \(\$ 10\) billion in \(2005 . \quad\) (Source: Christiansen Capital Advisors.) (a) Find a linear function given by $$ B(x)=m\left(x-x_{1}\right)+y_{1} $$ that models these data, where \(x\) is the year. (b) Use \(B(x)\) to estimate the years when consumer losses from online betting were more than \(\$ 6\) billion. Assume that the domain of \(B\) is 2002 to 2007 .

Step-by-Step Solution

Answer

Losses exceeded $6 billion after 2003, within the years 2004 to 2007.

Step 1: Identify Known Points

Given data points are 2002 with a loss of \(4 billion, and 2005 with a loss of \)10 billion. We identify the points as \(x_1, y_1\) = (2002, 4) and \(x_2, y_2\) = (2005, 10).

Step 2: Calculate Slope (m)

The slope \(m\) of the line is calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the known values, \(m = \frac{10 - 4}{2005 - 2002} = \frac{6}{3} = 2\).

Step 3: Write the Linear Function

Substitute \(m = 2\) and the point \(x_1, y_1\) = (2002, 4) into the point-slope form \(B(x) = m(x - x_1) + y_1\). Thus, \(B(x) = 2(x - 2002) + 4\).

Step 4: Find the Year When Losses Exceed $6 Billion

To find the year when losses exceed $6 billion, solve the inequality: \[B(x) > 6\] \Rightarrow 2(x - 2002) + 4 > 6\. Simplifying, \[2(x - 2002) > 2\] leads to \[x - 2002 > 1\] resulting in \[x > 2003\].

Step 5: Verify Domain and Find the Range

Given \(x\) ranges from 2002 to 2007, \(B(x)\) starts at \(B(2002) = 4\) and ends at \(B(2007) = 2(5) + 4 = 14\). Losses pass $6 billion between 2002 and 2007.

Key Concepts

Slope Calculation

The slope of a linear function represents how much the dependent variable changes for a unit change in the independent variable. In this context, the slope tells us how consumer gambling losses change over time. To calculate the slope, we use two points on the line. Here, the points are \((2002, 4)\) and \((2005, 10)\).

The formula for the slope \(m\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Substitute the values we have:

• \( m = \frac{10 - 4}{2005 - 2002} = \frac{6}{3} = 2 \)

This means that every year, consumer losses increased by \(2\) billion dollars. Understanding the slope is crucial for predicting future trends or assessing changes over a specific period.

A negative slope would indicate a decrease in losses over time, while a positive indicates growth, as we see in our example.

Point-Slope Form

The point-slope form is a method used to write the equation of a line when you know the slope and one point on the line. It is particularly useful in situations like this, where we need a linear model based on empirical data.

The point-slope form equation is:

• \( B(x) = m(x - x_1) + y_1 \)

In our exercise, we substitute \(m = 2\) and the point \((x_1, y_1) = (2002, 4)\) into the formula:

• \( B(x) = 2(x - 2002) + 4 \)

This equation models the relationship between the year \(x\) and consumer losses \(B(x)\).

It's a straightforward method that avoids the complexities of deriving a formula from scratch. Always remember that with a proper understanding of the initial data points, you can construct an accurate linear model to make predictions.

Inequality Solving

Solving inequalities is about finding the range of values that satisfy the conditions given. Here, we need to estimate when the consumer losses from online betting exceeded \(6\) billion dollars. To carry out this task, we substitute \(B(x)\) and solve for \(x\) in the inequality:

• \( B(x) > 6 \)
• \( 2(x - 2002) + 4 > 6 \)

Simplify the inequality:

• \( 2(x - 2002) > 2 \)
• \( x - 2002 > 1 \)

Therefore, \( x > 2003 \). This means that consumer losses exceeded \(6\) billion dollars starting from the year \(2004\).

Remember to respect the given domain, \(2002\) to \(2007\), ensuring your values make sense within the context of the problem. Inequality solving is essential in identifying key thresholds in various scenarios.