Data on airline revenue from​ add-on fees are listed in the table. Model the data with a linear function.​ Then, using that​ function, predict the revenue from airline​ add-on fees in 

20182018.

​Year, x

Airline Revenue from​ Add-On Fees​ (in billions), y
 
​2010, 0

​$

29.129.1

​2011, 1

30.830.8

​2012, 2

35.535.5

​2013, 3

40.740.7

​2014, 4

45.745.7

To model the data with a linear function, we will first determine the linear equation of the form:

\[ y = mx + b \]

Where:
- $ y $ is the airline revenue from add-on fees.
- $ x $ is the year since 2010 (i.e., $ x = 0 $ corresponds to 2010, $ x = 1 $ corresponds to 2011, and so on).
- $ m $ is the slope of the line.
- $ b $ is the y-intercept.

### Step 1: Calculate the slope ($ m $)

The slope $ m $ can be calculated using the formula:

\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]

Where:
- $ x_i $ and $ y_i $ are the data points.
- $ \bar{x} $ and $ \bar{y} $ are the averages of $ x $ and $ y $, respectively.

Let's calculate this step by step.

#### Data Points:
\[
\begin{array}{c|c}
x & y \text{ (in billions)} \\
\hline
0 & 29.1 \\
1 & 30.8 \\
2 & 35.5 \\
3 & 40.7 \\
4 & 45.7 \\
\end{array}
\]

### Step 2: Find the average of $ x $ and $ y $

\[
\bar{x} = \frac{0 + 1 + 2 + 3 + 4}{5} = 2
\]
\[
\bar{y} = \frac{29.1 + 30.8 + 35.5 + 40.7 + 45.7}{5} = 36.36
\]

### Step 3: Calculate $ m $ (slope)

\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{(0-2)(29.1-36.36) + (1-2)(30.8-36.36) + (2-2)(35.5-36.36) + (3-2)(40.7-36.36) + (4-2)(45.7-36.36)}{(0-2)^2 + (1-2)^2 + (2-2)^2 + (3-2)^2 + (4-2)^2}
\]

Let's calculate these sums and get the slope and y-intercept.The linear function that models the data is:

\[
y = 4.31x + 27.74
\]

### Step 4: Predict the revenue for the year 2018 (which corresponds to $ x = 8 $ since 2010 is $ x = 0 $).

\[
y_{2018} = 4.31(8) + 27.74
\]

Let's calculate this.The predicted revenue from airline add-on fees in 2018 is approximately **$62.22 billion**.

Do you want more details on how this was calculated, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How does the linear model's prediction compare to actual revenue data from 2018?
2. What factors could cause deviations from the linear model in real-world scenarios?
3. How would this model change if we used data beyond 2014?
4. Can we use a different type of function (e.g., quadratic) to model this data more accurately?
5. What are the limitations of using a linear model for predicting future revenue?

**Tip:** Always check the residuals of your model to ensure that a linear model is the best fit for the data.

Learn how to model airline revenue from add-on fees using a linear function. Calculate the slope and y-intercept, and predict revenue for a given year. This problem covers linear regression concepts and is suitable for high school students.

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