Question

Find a quadratic equation that models the decline in physical recorded music revenue​ (not digital). Let x number of years since 2000 and y be the physical recorded music revenue in millions of dollars. Use this model to fill in the predicted physical revenue below and compare with the actual given data.

Answer 1

The rate of decrease in the revenue reduces each year, resulting in an

approximate quadratic equation model that is concave upwards.

$\mathrm{The \ quadratic \ equation \ models \ is}; \ \underline{y = 76.5 \cdot x^2 - 2264.8 \overline 3 \cdot x + 18769. \overline 6}$

The difference between the actual and predicted values are;

$\begin{array}{|c|c|c|c|}\underline{Year}& \underline{Difference \ in \ Values}\\2008&0\\2009&-143.\overline 6\\2010&-118.\overline 3\\2011&0\\2012& 62.\overline 3\\2013&71.\overline 6\\2014&0\end{array}\right]$

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

$a = \dfrac{1377}{18} = 76.5$

a = 76.5

From, equation (4), 2434 =  -57·a - 3·b, we have;

2434 =  -57 × 76.5  - 3·b

Therefore;

$b = \dfrac{2434 + 57 \times 76.5}{-3} = -\dfrac{13589}{6} = -2264.8 \overline{3}$

b = -2264.8$\mathbf{\overline 3}$

From equation (1), we get;

5547 = 64 × 76.5 + 8×(-2264.8$\overline 3$) + c

Therefore;

c = 5547 - (64 × 76.5 + 8×(-2264.8$\overline 3$)) = 18769.$\overline 6$

c = 18769.$\mathbf{\overline 6}$

Therefore, the quadratic model is; $\underline{y = 76.5 \cdot x^2 - 2264.8 \overline 3 \cdot x + 18769. \overline 6}$

Verifying, we have in year 2012, x = 12

y = 76.5×12² - 2264.8$\overline 3$×12 + 18769.$\overline 6$ = 2607.74

The completed table and comparison of the predicted physical revenue

and actual values is presented as follows;

$\begin{array}{|c|c|c|c|}Year&Actual \ Revenue&Predicted \ Revenue& Difference \\2008&5547&5547&0\\2009&4439&4582.\overline 6&-143.\overline 6\\2010&3653&3771.\overline 3&-118.\overline 3\\2011&3113&3113&0\\2012&2670&2607. \overline 6 & 62.\overline 3\\2013&2327&2255.\overline 3&71.\overline 6\\2014&2056&2056&0\end{array}\right]$

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