Problem 31 - A Graphical Approach to College Algebra

Question

Solve each problem. Online Holiday Shopping In 2011 , online holiday sales were \(\$ 192\) billion, and in 2014 , they were \(\$ 249\) billion. (Source: Digital Lifestyles.) (a) Find a linear function \(S\) that models these data, where \(x\) is the year. (b) Interpret the slope of the graph of \(S\). (c) Predict when online holiday sales might reach \(\$ 325\) billion.

Step-by-Step Solution

Answer

The linear function is \( S(x) = 19x - 38059 \). The slope of 19 indicates sales increase by $19 billion annually. Sales might reach $325 billion in 2021.

Step 1: Understand the Problem

We are given the online holiday sales for two different years (2011 and 2014) and need to find a linear function that models the data. We will also interpret the slope of this function and use it to predict when sales will reach $325 billion.

Step 2: Define the Variables

Let \( S(x) \) represent the sales in billions of dollars, and \( x \) represent the year. We will use the given data points: \((2011, 192)\) and \((2014, 249)\) to find a linear function.

Step 3: Find the Slope

The slope \( m \) can be found using the formula:\[ m = \frac{S(2014) - S(2011)}{2014 - 2011} = \frac{249 - 192}{3} = \frac{57}{3} = 19. \] This means sales increased by $19 billion per year.

Step 4: Find the Equation of the Line

Use the point-slope form of a line equation \( y - y_1 = m(x - x_1) \). Using the point \((2011, 192)\) and slope \(m = 19\), we have:\[ S(x) - 192 = 19(x - 2011). \] Simplify to get the function: \[ S(x) = 19x - 38059. \]

Step 5: Interpret the Slope

The slope of 19 indicates that online holiday sales are increasing by $19 billion per year.

Step 6: Predict when Sales reach $325 billion

Set \(S(x) = 325\) and solve for \(x\): \[ 325 = 19x - 38059. \] Add 38059 to both sides: \[ 325 + 38059 = 19x, \] resulting in \[ 38384 = 19x. \] Divide both sides by 19: \[ x = \frac{38384}{19} \approx 2020.74. \] Sales are projected to reach $325 billion around the year 2021.

Key Concepts

Slope

The slope plays a fundamental role in understanding linear functions. In this problem, the slope tells us how much online holiday sales change each year. Calculating the slope involves determining the rate at which a quantity changes. You can find it using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]In our example, we calculated the slope using the values for the years 2011 and 2014, where sales were \(192 billion and \)249 billion respectively. Plug these values into our formula:- Numerator: Difference in sales: \(249 - 192 = 57\)- Denominator: Difference in years: \(2014 - 2011 = 3\)Then, divide to find the slope: \(m = \frac{57}{3} = 19\). This tells us that online holiday sales increased by $19 billion each year between 2011 and 2014.

Data Modeling

Data modeling involves creating a mathematical representation of real-world processes and helps in understanding trends. Linear functions like the one in our problem offer a simple way to model relationships when data suggests a constant rate of change.To construct a linear model, we need to determine the equation of the line that fits the given points. This equation takes the form of \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. We used the slope calculated previously and plugged in the data point for 2011:- Slope \(m = 19\)- Point: \((2011, 192)\)Apply the point-slope formula:\[S(x) - 192 = 19(x - 2011)\]Simplify it to get the model:\[S(x) = 19x - 38059\]Data modeling like this allows us to predict future trends and understand past behavior efficiently.

Equation of a Line

An equation of a line characterizes a line in terms of its slope and y-intercept in a two-dimensional space using the formula \(y = mx + b\). Here, \(y\) is the dependent variable, \(m\) is the slope, \(x\) is the independent variable, and \(b\) is the y-intercept.In our exercise, we have:- Slope \(m = 19\)- Using the point \((2011, 192)\), the equation becomes:\[S(x) = 19x - 38059\]This equation tells us that sales \(S(x)\) depend linearly on the year \(x\). Calculating the y-intercept involves solving \(b\) by substituting known values into the line's equation. It indicates the start value of our data model when \(x = 0\), which might not always make practical sense in context but serves mathematical purposes in defining the line.

Sales Prediction

Sales prediction uses the linear model to forecast future trends. With the determined equation \(S(x) = 19x - 38059\), we can predict when sales will reach a certain value, such as \(325 billion.This involves solving for \(x\) when \(S(x) = 325\):- Begin by substituting into the equation:\[325 = 19x - 38059\]- Add 38059 to both sides:\[38384 = 19x\]- Divide by 19 to solve for \(x\):\[x = \frac{38384}{19} \approx 2020.74\]This approximation suggests that online holiday sales might reach \)325 billion in the year 2021. This predictive power of linear models helps businesses and analysts make data-driven decisions.