Retail e-commerce sales in Germany (excluding event tickets, financial products, and travel) are shown below.

**German e-Commerce**

YearInternet Sales
(billion euros)

2003

5.8

2004

7.6

2005

9.0

2006

10.2

2007

11.4

2008

13.6

(a) Why is a cubic model appropriate for the data?


The scatter plot is increasing and appears to be concave up before changing to concave down.The scatter plot is decreasing and appears to be concave up before changing to concave down.    The scatter plot is increasing and appears to be concave down before changing to concave up.The scatter plot is decreasing and appears to be concave down before changing to concave up.


(b) Align the input to years since 2000. Write the function of the cubic model that gives the retail e-commerce sales in Germany in billion euros, where *x* is the number of years since 2000, with data from 

3 ≤ *x* ≤ 8.

 (Round all numerical values to three decimal places.)


*s*(*x*) = 
 


(c) Write the input value for the approximate location of the inflection point.


*x* = 

Let's break down each part of the problem to address the questions one by one.

### (a) Why is a cubic model appropriate for the data?

A cubic model is appropriate when the scatter plot exhibits changes in concavity (i.e., the curve changes from concave up to concave down or vice versa). In this case, the retail e-commerce sales data in Germany shows an increasing trend, but the rate of increase starts to slow down after a certain point, indicating a change in concavity.

The description states:
- **The scatter plot is increasing and appears to be concave up before changing to concave down.**

This matches the characteristics of a cubic function, which typically shows an inflection point where the concavity changes. Therefore, a cubic model is suitable for modeling this data because of the observed changes in the growth trend.

### (b) Align the input to years since 2000. Write the function of the cubic model that gives the retail e-commerce sales in Germany in billion euros, where \(x\) is the number of years since 2000, with data from \(3 \leq x \leq 8\).

First, let's redefine the years as the number of years since 2000. So the new inputs are:

\[
\begin{aligned}
\text{Year} & : 2003, 2004, 2005, 2006, 2007, 2008 \\
x & : 3, 4, 5, 6, 7, 8 \\
\text{Sales} & : 5.8, 7.6, 9.0, 10.2, 11.4, 13.6 \ (\text{billion euros})
\end{aligned}
\]

To construct a cubic model, we assume the sales function \( s(x) \) can be expressed as:

\[
s(x) = ax^3 + bx^2 + cx + d
\]

We can use the data points \( (3, 5.8), (4, 7.6), (5, 9.0), (6, 10.2), (7, 11.4), (8, 13.6) \) to set up a system of equations to determine the coefficients \(a\), \(b\), \(c\), and \(d\). For simplicity, I will calculate the cubic regression model for you.

### (c) Write the input value for the approximate location of the inflection point.

The inflection point of a cubic function occurs where the second derivative changes sign, which is where the concavity changes. To find the inflection point for \( s(x) = ax^3 + bx^2 + cx + d \), we first take the second derivative of \(s(x)\):

\[
s'(x) = 3ax^2 + 2bx + c
\]
\[
s''(x) = 6ax + 2b
\]

Setting \( s''(x) = 0 \) to find the inflection point:

\[
6ax + 2b = 0
\]
\[
x = -\frac{b}{3a}
\]

Once we have the values for \(a\) and \(b\), we can compute the approximate value of \(x\) at the inflection point.

I'll calculate both parts (b) and (c) now.### (b) The cubic model for retail e-commerce sales in Germany is:

\[
s(x) = 0.070x^3 - 1.140x^2 + 7.269x - 7.676
\]
where \(x\) is the number of years since 2000, and \(x\) ranges from 3 to 8.

### (c) The approximate location of the inflection point occurs at:

\[
x \approx 5.398
\]

This means the inflection point occurs approximately 5.4 years after 2000, which corresponds to a time around mid-2005.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How would extending the data range affect the cubic model?
2. Can a quadratic model fit the data reasonably well? Why or why not?
3. How does the location of the inflection point affect future trends in sales growth?
4. What other functions might model this data if there were more fluctuations in the future?
5. What is the significance of the coefficients in the context of sales growth?

**Tip**: The inflection point in a cubic model is crucial because it marks the transition between accelerating and decelerating growth.

Retail e-commerce sales in Germany (excluding event tickets, financial products, and travel) are shown below.

**German e-Commerce**

Year | Internet Sales (billion euros)
---|---
2003 | 5.8
2004 | 7.6
2005 | 9.0
2006 | 10.2
2007 | 11.4
2008 | 13.6

(a) Why is a cubic model appropriate for the data?

(b) Align the input to years since 2000. Write the function of the cubic model that gives the retail e-commerce sales in Germany in billion euros, where x is the number of years since 2000, with data from 3 ≤ x ≤ 8.

(c) Write the input value for the approximate location of the inflection point.

Explore the use of cubic models to analyze retail e-commerce sales in Germany from 2003 to 2008. Learn how to fit a cubic function to the data and calculate the inflection point, providing insights into sales growth trends. Suitable for students in Grades 11-12.

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