Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Holiday Shopping In \(2012,\) U.S. holiday sales were \(\$ 569\) billion, and in 2015 , they were \(\$ 626\) billion. (Source: National Retail Federation.) (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 the year that U.S. holiday sales might reach \(\$ 721\) billion.

Step-by-Step Solution

Answer

(a) \(S(x) = 19x - 37700\). (b) The sales increase by \$19 billion per year. (c) Year 2021.

Step 1: Define Points from Data

Identify the known data points from the years 2012 and 2015. These points can be represented as \((2012, 569)\) and \((2015, 626)\) where the first component is the year and the second is the sales in billions.

Step 2: Calculate the Slope

Use the two data points to calculate the slope \(m\) using the formula for the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given values, we have \(m = \frac{626 - 569}{2015 - 2012} = \frac{57}{3} = 19\). The slope is 19.

Step 3: Form the Linear Function

With the slope \(m = 19\) and the point-slope form \(y - y_1 = m(x - x_1)\), we use one of the points, say \((2012, 569)\), to find the equation of the line: \(y - 569 = 19(x - 2012)\). Simplifying, we get \(y = 19x - 38269 + 569\), which simplifies to \(y = 19x - 37700\). Therefore, the linear function is \(S(x) = 19x - 37700\).

Step 4: Interpret the Slope

The slope of the line, \(19\), indicates the average increase in holiday sales per year, in billions of dollars. This means the holiday sales increased by approximately \$19 billion each year from 2012 to 2015.

Step 5: Predict the Year for Sales of $721 Billion

Set the function \(S(x) = 721\) and solve for \(x\). Using the equation \(721 = 19x - 37700\), add \(37700\) to both sides to get \(721 + 37700 = 19x\). Simplifying, we find \(38421 = 19x\). Dividing both sides by \(19\), we have \(x = \frac{38421}{19} \approx 2021\). Thus, holiday sales might reach \$721 billion in the year 2021.

Key Concepts

Slope Calculation

When talking about linear functions, one of the most essential concepts to understand is the slope. The slope is a number that describes how steep a line is. It tells us how much the dependent variable changes when the independent variable increases by one.To calculate the slope between two points, we use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \(y_2\) and \(y_1\) are the y-values (in this case, sales), and \(x_2\) and \(x_1\) are the x-values (years). In our exercise, the points are \( (2012, 569) \) and \( (2015, 626) \), representing sales in billions. Substituting these values gives us:\[ m = \frac{626 - 569}{2015 - 2012} = \frac{57}{3} = 19 \]Thus, the slope is 19. This means that each year, on average, holiday sales increased by 19 billion dollars.

Point-Slope Form

Once we know the slope, the next step is to form the linear equation, which helps predict future values. The point-slope form is one of the most useful ways to write the equation of a line. The formula is:\[ y - y_1 = m(x - x_1) \]Here, \(m\) is the slope, and \( (x_1, y_1) \) is a point on the line. For our problem, we used the slope \(m = 19\) and the point \( (2012, 569) \) to form the equation:\[ y - 569 = 19(x - 2012) \]By simplifying, we get:\[ y = 19x - 38269 + 569 \ y = 19x - 37700\]So, the linear function that models the sales is \( S(x) = 19x - 37700 \). Now, anyone can use this function to estimate sales for different years.

Data Interpretation

Understanding what the slope and the linear function tell us is crucial for interpreting data effectively.- **Slope Interpretation**: Here, the slope of 19 indicates that from 2012 to 2015, the U.S. holiday sales increased, on average, by 19 billion dollars each year. This upward trend is a good indicator of economic growth during that period.- **Predicting Future Values**: By setting \( S(x) = 721 \), we solve for \( x \) to predict when sales will hit 721 billion: \[ 721 = 19x - 37700 \ 38421 = 19x \ x = \frac{38421}{19} \approx 2021 \] From the calculation above, we predict that sales could reach 721 billion by the year 2021. This is an example of how linear functions can help in making future predictions based on historical data.