Solve each problem. (Modeling) Online Retail Sales Projected retail e-commerce sales (in billions of dollars) for the years 2016–2022 can be modeled by the equation y=52.304x+396.80, where x=0 corresponds to 2016, x=1 corresponds to 2017, and so on. Based on this model, find projected retail e-commerce sales in 2022 to the nearest tenth of a billion. (Data from www.statista.com)

Verified step by step guidance

1. Identify the given linear model for projected retail e-commerce sales: \(y = 52.304x + 396.80\), where \(x\) represents the number of years since 2016.
2. Determine the value of \(x\) that corresponds to the year 2022. Since \(x=0\) is 2016, calculate \(x\) as \$2022 - 2016$.
3. Substitute the value of \(x\) found in step 2 into the equation \(y = 52.304x + 396.80\) to find the projected sales for 2022.
4. Perform the multiplication and addition operations in the equation to express \(y\) in terms of billions of dollars.
5. Round the resulting value of \(y\) to the nearest tenth of a billion to get the final projected retail e-commerce sales for 2022.

Key Concepts

Linear Functions and Models

A linear function models a relationship between two variables with a constant rate of change, represented by y = mx + b. Here, y represents sales, x represents years since 2016, m is the slope indicating yearly increase, and b is the initial value at x=0. Understanding this helps interpret and use the equation to predict values.

Interpreting Variables and Parameters

In the given model, x corresponds to the number of years after 2016, and y represents projected sales in billions. Recognizing how variables relate to real-world quantities is essential for correctly substituting values and interpreting results, such as identifying x=6 for the year 2022.

Evaluating Functions

Evaluating a function means substituting a specific input value into the equation to find the output. For this problem, substituting x=6 into y=52.304x+396.80 calculates the projected sales for 2022. This process is fundamental for applying models to make predictions.