To find a quadratic equation that models the decline in physical recorded music revenue, we consider a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( x \)[/tex] is the number of years since 2000, and [tex]\( y \)[/tex] represents the physical recorded music revenue in millions of dollars.
Step 1: Setting up the quadratic model
We'll find the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] that best fit the given data:
- Actual revenues: [5535, 4440, 3665, 3120, 2678, 2326, 2047]
- Corresponding years: [2008, 2009, 2010, 2011, 2012, 2013, 2014]
We convert these years into [tex]\( x \)[/tex] values (number of years since 2000):
- [tex]\( x \)[/tex] values: [8, 9, 10, 11, 12, 13, 14]
By fitting a quadratic model to the data, we obtain coefficients:
[tex]\[ a = 76.2024, \, b = -2236.4167, \, c = 18476.8571 \][/tex]
So, the quadratic equation is:
[tex]\[ y = 76.2024x^2 - 2236.4167x + 18476.8571 \][/tex]
Step 2: Predicting the physical revenue using the model
Using the quadratic equation [tex]\( y = 76.2024x^2 - 2236.4167x + 18476.8571 \)[/tex], we predict the physical recorded music revenue for the years 2008 to 2014.
- For 2008 ([tex]\( x = 8 \)[/tex]):
[tex]\[ y = 76.2024(8^2) - 2236.4167(8) + 18476.8571 = 5462.48 \][/tex]
- For 2009 ([tex]\( x = 9 \)[/tex]):
[tex]\[ y = 76.2024(9^2) - 2236.4167(9) + 18476.8571 = 4521.50 \][/tex]
- For 2010 ([tex]\( x = 10 \)[/tex]):
[tex]\[ y = 76.2024(10^2) - 2236.4167(10) + 18476.8571 = 3732.93 \][/tex]
- For 2011 ([tex]\( x = 11 \)[/tex]):
[tex]\[ y = 76.2024(11^2) - 2236.4167(11) + 18476.8571 = 3096.76 \][/tex]
- For 2012 ([tex]\( x = 12 \)[/tex]):
[tex]\[ y = 76.2024(12^2) - 2236.4167(12) + 18476.8571 = 2613.00 \][/tex]
- For 2013 ([tex]\( x = 13 \)[/tex]):
[tex]\[ y = 76.2024(13^2) - 2236.4167(13) + 18476.8571 = 2281.64 \][/tex]
- For 2014 ([tex]\( x = 14 \)[/tex]):
[tex]\[ y = 76.2024(14^2) - 2236.4167(14) + 18476.8571 = 2102.69 \][/tex]
Step 3: Comparing predicted and actual revenues
Finally, we compare the predicted revenues with the actual given data.
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Year} & \text{Actual Revenue} & \text{Predicted Revenue} \\
\hline
2008 & 5535 & 5462.48 \\
2009 & 4440 & 4521.50 \\
2010 & 3665 & 3732.93 \\
2011 & 3120 & 3096.76 \\
2012 & 2678 & 2613.00 \\
2013 & 2326 & 2281.64 \\
2014 & 2047 & 2102.69 \\
\hline
\end{array}
\][/tex]
The predicted values are close to the actual values, indicating that the quadratic model fits the data well and can reasonably predict the trend in the decline of physical recorded music revenue.
