Digital Music Revenues The total projected worldwide digital music revenues R, in millions of dollars, for the years $2005$through $2010$can be estimated by the function.

$R\left(x\right)=170.7x^2+1373x+1080$

where $x$is the number of years after $2005$.

$\left(a\right)$Find $R\left(0\right)$, $R\left(3\right)$, and $R\left(5\right)$and explain what each value represents.

$\left(b\right)$Find $r=R(x-5)$.

$\left(c\right)$Find $r\left(5\right)$, $r\left(8\right)$, and $r\left(10\right)$and explain what each value represents.

$\left(d\right)$In the model $r$, what does $x$represent?

$\left(e\right)$Would there be an advantage in using the model $r$when estimating the projected revenues for a given year instead of the model $R$?

$R\left(x\right)=170.7x^2+1373x+1080$

Substitute $x=0$in the given function.

$$\begin{gathered} R\left(0\right)=170.7\left(0\right)+1373\left(0\right)+1080 \\ R\left(0\right)=1080 \end{gathered}$$

where $x$is the number of years after $2005$.

we can say that, $R\left(0\right)$represents the revenue for the year $2005$.

Substitute $x=3$in the given function.

$$\begin{gathered} R\left(3\right)=170.7{\left(3\right)}^2+1373\left(3\right)+1080 \\ R\left(3\right)=6735.3 \end{gathered}$$

we can say that, $R\left(3\right)$represents the revenue for the year $2008$.

Substitute $x=5$in the given function.

$$\begin{gathered} R\left(5\right)=170.7{\left(5\right)}^2+1373\left(5\right)+1080 \\ R\left(5\right)=12212.5 \end{gathered}$$

we can say that, $R\left(5\right)$represents the revenue for the year$2010$.

$R(x-5)=170.7{\left(x-5\right)}^2+1373\left(x-5\right)+1080$

So, the obtained equation is,

$r==170.7{\left(x-5\right)}^2+1373\left(x-5\right)+1080$

$$\begin{gathered} r\left(5\right)=107.7{\left(5-5\right)}^2+1373\left(5-5\right)+1080 \\ r\left(5\right)=1080 \end{gathered}$$

we can say that, $r\left(5\right)$represents the revenue for the year $2005$.

Substitute $x=8$in the equation.

$$\begin{gathered} r\left(8\right)=170.7{\left(8-5\right)}^2+1373\left(8-5\right)+1080 \\ r\left(8\right)=170.7{\left(3\right)}^2+1373\left(3\right)+1080 \\ r\left(8\right)=6735.3 \end{gathered}$$

we can say that, $r\left(8\right)$represents the revenue for the year $2008$.

Substitute $x=10$in the equation.

$$\begin{gathered} r\left(10\right)=170.7{\left(10-5\right)}^2+1373\left(10-5\right)+1080 \\ r\left(10\right)=170.7{\left(5\right)}^2+1373\left(5\right)+1080 \\ r\left(10\right)=12212.5 \end{gathered}$$

we can say that,$r\left(10\right)$represents the revenue for the yearrole="math" localid="1645955353322" $2005$.

In the model$r$,$x$represents the number of years after$2000$.

The model $R$is used to determine the total projected world wide digital music revenue for the years after $2005$, where as when the model $r$is used we can determine the revenue from the year $2000$.