https://file.aiquickdraw.com/mathbot/file/1726709468978zf3z6kj6.jpg

To solve the given problem from the image, let's break it down step by step:

### Part (a): Find a quadratic regression model for the data.

The table shows the product's sales over five years (2005-2009). The corresponding years (denoted by \( t \)) are:

- Year \( t = 0 \): Sales \( S = 22.3 \) million
- Year \( t = 1 \): Sales \( S = 39.4 \) million
- Year \( t = 2 \): Sales \( S = 51.7 \) million
- Year \( t = 3 \): Sales \( S = 54.8 \) million
- Year \( t = 4 \): Sales \( S = 54.9 \) million

We need to fit a quadratic regression model \( S(t) = at^2 + bt + c \) using these data points.

I will calculate the quadratic regression for this data now.The quadratic regression model for the sales data is:

\[
S(t) = -3.086t^2 + 20.403t + 22.329
\]

Where:
- \( a = -3.086 \)
- \( b = 20.403 \)
- \( c = 22.329 \)

### Part (b): Predict the product's sales for the years 2010 and 2011

For 2010, \( t = 5 \) (since \( t = 0 \) corresponds to 2005), and for 2011, \( t = 6 \). We will use the quadratic model to predict the sales for these years.

Let me calculate these predictions.The predicted sales for the product are:

- In 2010 (\( t = 5 \)): **47.2 million** dollars
- In 2011 (\( t = 6 \)): **33.7 million** dollars

Would you like any further details or clarifications?

Here are 5 related questions to expand on this problem:
1. How can we calculate a cubic regression model for the same data?
2. What would be the predicted sales for the year 2012 using this quadratic model?
3. How does the shape of the graph change when using a quadratic model compared to a linear one?
4. Can this quadratic model be used for long-term sales predictions?
5. How would outliers affect the accuracy of the regression model?

**Tip**: When using regression models, always check if a higher-degree polynomial (like cubic) provides a better fit for your data.

The following table shows a tech product's sales during the financial years 2005–2009 (t is time in years since 2005). Use the data to: (a) Find a quadratic regression model. (b) Predict the product's sales for 2010 and 2011.

Learn how to find a quadratic regression model for tech product sales data between 2005 and 2009. This problem covers key algebraic concepts, including quadratic regression using real-world data and predictions for future sales in 2010 and 2011.

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