https://file.aiquickdraw.com/mathbot/file/17261465866264zh0fys7.jpg

Let's break down the questions from the image:

1. **Part (a)** asks for the sales in 2021. From the table:
   - Sales in 2021 were **62** thousand dollars.

2. **Part (b)** asks for the yearly increase in sales. To find the average yearly increase, we can calculate the differences between consecutive years:
   - 2019: \( 34 - 20 = 14 \)
   - 2020: \( 48 - 34 = 14 \)
   - 2021: \( 62 - 48 = 14 \)
   - 2022: \( 76 - 62 = 14 \)

   The increase in sales is consistent at **14 thousand** dollars each year.

3. **Part (c)** asks for a linear function to model the sales data, with \( x = 1 \) corresponding to 2018, and \( S(x) \) being the sales in thousands.
   - We can model this with the equation \( S(x) = mx + b \), where \( m \) is the slope (the yearly increase) and \( b \) is the initial sales in 2018.
   - The slope \( m = 14 \) (since sales increase by 14 thousand each year).
   - For \( x = 1 \) (year 2018), \( S(1) = 20 \), so we can substitute into the equation:
     \[
     20 = 14(1) + b \quad \Rightarrow \quad b = 6
     \]
   - Therefore, the linear function is: \( S(x) = 14x + 6 \).

4. **Part (d)** asks to predict the sales for 2023 using the function. 
   - For 2023, \( x = 6 \) (since \( x = 1 \) for 2018, \( x = 6 \) for 2023).
   \[
   S(6) = 14(6) + 6 = 84 + 6 = 90
   \]
   The predicted sales for 2023 is **90 thousand** dollars.

Let me know if you need further clarification or details on these steps.

Here are 5 related questions to expand your understanding:
1. How can you verify the consistency of the yearly increase using a different method?
2. What would the sales have been in 2017 using the linear model?
3. If the sales growth changed after 2022, how would you modify the model?
4. Can you explain how the slope impacts the rate of sales growth?
5. How does the initial value \( b \) in the linear model influence predictions for earlier years?

**Tip**: When creating linear models, always check that the slope correctly represents the change across all points in the dataset for accuracy.

The table shows total dollar sales (in thousands) for a local retailer for various years. 
Year: 2018 - Sales: 20, 2019 - Sales: 34, 2020 - Sales: 48, 2021 - Sales: 62, 2022 - Sales: 76. 

(a) What were the sales for 2021?
(b) By how much did sales increase each year?
(c) Find a linear function that models the above data. Let x = 1 correspond to 2018 and let S(x) be in thousands.
(d) Use S(x) to predict sales in 2023.

Learn how to model the growth of sales from 2018 to 2022 using a linear function. This problem explores the calculation of yearly increases, prediction of future sales, and constructing a linear equation based on data points. Perfect for students in grades 9-10.

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