Chapter 6.8, Problem 43PPS

a.

To determine

To write a polynomial equation to find the year for the given music sales.

Answer to Problem 43PPS

  $30x^3-478x^2+1,758x+1,092=0$

Explanation of Solution

Given:

The music sales will be about $\$20,000,000,000$

Calculation:

The given function for the problem is

  $f(x)=30x^3-478x^2+1,758x+10,092$

The value of f(x) would be $9000$ . This is because $9,000,000,000$ will be divided by one million as f ( x ) shown in millions.

Substitute this to the function to get a polynomial equation as follows-

  $\begin{array}{l}f(x)=30x^3-478x^2+1,758x+10,092\\ 9000=30x^3-478x^2+1,758x+10,092\\ 0=30x^3-478x^2+1,758x+10,092-9000\\ 30x^3-478x^2+1,758x+1,092=0\end{array}$

b.

To determine

To list the possible whole number of solutions for the given equation in part (a).

Answer to Problem 43PPS

  $1,2,3,4,6,7,12,13,14,21,26,28,39,42,52,78,84,91,156,182,273,364,546,1092$

Explanation of Solution

The following is the equation −

  $30x^3-478x^2+1,758x+1,092=0$

The possible whole number solutions of $1092$ are as follows-

  $1,2,3,4,6,7,12,13,14,21,26,28,39,42,52,78,84,91,156,182,273,364,546,1092$

c.

To determine

To find the year in which music sales will reach $\$20,000,000,000$

Answer to Problem 43PPS

  $2013-2014$ .

Explanation of Solution

Given:

The music sales will reach $\$20,000,000,000$

Calculation:

To know the year just subsitute any values from the possible ones in the following equation as follws-

  $\begin{array}{l}30x^3-478x^2+1,758x+10,092=9000\\ \text{on substituting }x=8\\ $ 30{(8)}^3-478{(8)}^2+1,758(8)+10,092=9000$8924<9000\\ \text{on substituting }x=9\\ $ 30{(9)}^3-478{(9)}^2+1,758(9)+10,092=9000$9066>9000\end{array}$

The year that will reach the target sales will be somewhere 2013 and 2014 since $2005+8=2013\text{ and }2005+9=2014$

d.

To determine

To find whether the model represent a realistic estimate for all future music sales.

Answer to Problem 43PPS

No.

Explanation of Solution

The model and function that can be used to estimate the music sales is not a realistic estimation for all future music sales. This is because as years go by, there are so many factors that could trigger the increase or decrease in sales. This is not accurate and could not be a basis for future estimation.