The rate of decrease in the revenue reduces each year, resulting in an
approximate quadratic equation model that is concave upwards.
[tex]\mathrm{The \ quadratic \ equation \ models \ is}; \ \underline{y = 76.5 \cdot x^2 - 2264.8 \overline 3 \cdot x + 18769. \overline 6}[/tex]
The difference between the actual and predicted values are;
[tex]\begin{array}{|c|c|c|c|}\underline{Year}& \underline{Difference \ in \ Values}\\2008&0\\2009&-143.\overline 6\\2010&-118.\overline 3\\2011&0\\2012& 62.\overline 3\\2013&71.\overline 6\\2014&0\end{array}\right][/tex]
Reasons:
The variables are;
x = Number of years since 2,000
y = Revenue for physically recorded music in millions of dollars
Required:
To model the decline using a quadratic equation.
Solution:
The general form of the quadratic equation is; y = a·x² + b·x + c
When y = 5547, x = 8
We get;
5547 = a·8² + b·8 + c = 64·a + 8·b + c
5547 = 64·a + 8·b + c...(1)
When y = 3113, x = 11
We get;
3113 = a·11² + b·11 + c = 121·a + 11·b + c
3113 = 121·a + 11·b + c...(2)
When y = 2056, x = 14
We get;
2056 = a·14² + b·14 + c = 196·a + 14·b + c
2056 = 196·a + 14·b + c...(3)
Subtracting equation (2) from equation (1) gives;
5547 - 3113 = 64·a + 8·b + c - (121·a + 11·b + c) = -57·a - 3·b
2434 = -57·a - 3·b...(4)
Subtracting equation (3) from equation (2) gives;
3113 - 2056 = 121·a + 11·b + c - (196·a + 14·b + c) = -75·a - 3·b
1057 = -75·a - 3·b...(5)
Subtracting equation (5) from equation (4) gives;
2434 - 1057 = -57·a - 3·b - (-75·a - 3·b) = 18·a
1377 = 18·a
[tex]a = \dfrac{1377}{18} = 76.5[/tex]
a = 76.5
From, equation (4), 2434 = -57·a - 3·b, we have;
2434 = -57 × 76.5 - 3·b
Therefore;
[tex]b = \dfrac{2434 + 57 \times 76.5}{-3} = -\dfrac{13589}{6} = -2264.8 \overline{3}[/tex]
b = -2264.8[tex]\mathbf{\overline 3}[/tex]
From equation (1), we get;
5547 = 64 × 76.5 + 8×(-2264.8[tex]\overline 3[/tex]) + c
Therefore;
c = 5547 - (64 × 76.5 + 8×(-2264.8[tex]\overline 3[/tex])) = 18769.[tex]\overline 6[/tex]
c = 18769.[tex]\mathbf{\overline 6}[/tex]
Therefore, the quadratic model is; [tex]\underline{y = 76.5 \cdot x^2 - 2264.8 \overline 3 \cdot x + 18769. \overline 6}[/tex]
Verifying, we have in year 2012, x = 12
y = 76.5×12² - 2264.8[tex]\overline 3[/tex]×12 + 18769.[tex]\overline 6[/tex] = 2607.74
The completed table and comparison of the predicted physical revenue
and actual values is presented as follows;
[tex]\begin{array}{|c|c|c|c|}Year&Actual \ Revenue&Predicted \ Revenue& Difference \\2008&5547&5547&0\\2009&4439&4582.\overline 6&-143.\overline 6\\2010&3653&3771.\overline 3&-118.\overline 3\\2011&3113&3113&0\\2012&2670&2607. \overline 6 & 62.\overline 3\\2013&2327&2255.\overline 3&71.\overline 6\\2014&2056&2056&0\end{array}\right][/tex]
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https://brainly.com/question/11826603
