The rate of decrease in the revenue reduces each year, resulting in an
approximate quadratic equation model that is concave upwards.
The difference between the actual and predicted values are;
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 = 76.5
From, equation (4), 2434 = -57·a - 3·b, we have;
2434 = -57 × 76.5 - 3·b
Therefore;
b = -2264.8
From equation (1), we get;
5547 = 64 × 76.5 + 8×(-2264.8) + c
Therefore;
c = 5547 - (64 × 76.5 + 8×(-2264.8)) = 18769.
c = 18769.
Therefore, the quadratic model is;
Verifying, we have in year 2012, x = 12
y = 76.5×12² - 2264.8×12 + 18769.
= 2607.74
The completed table and comparison of the predicted physical revenue
and actual values is presented as follows;
Learn more here:
