Q. 86 - Precalculus Enhanced with Graphing Utilities

Question

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 $R$?

Step-by-Step Solution

Answer

$\left(a\right) R\left(0\right)=1080$

This value represents the revenue for the year $2005$.

$R\left(3\right)=6735.3$

This value represents the revenue for the year $2008$

$R\left(5\right)=12212.5$

this value represents the revenue after $2010$.

$\left(b\right)R\left(x-5\right)=170.7\left(x-5\right)+1373\left(x-5\right)+1080$

$\left(c\right)r\left(5\right)=1080$

This value represents the revenue for the year $2005$.

$r\left(8\right)=6735.3$

This value represents the revenue for the year $2008$.

$r\left(10\right)=12212.5$

This value represents the revenue for the year $2005$.

$\left(d\right)$In the model $r, x$ represents the number of years after $2000$.

$\left(e\right)$ 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 $2000$.

Part a Step 1 Given function is,

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

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 $2010$.

Part b Step 1 Substitute x = x - 5 in the given function.

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

So, the obtained equation is,

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

Part c Step 1 Substitute x = 5 in the equation obtained from the part b .

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

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 year  $2005$.

Part d Step 1 The representation of x in the model r :

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

Part e Step 1 The advantage of the model r .

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 $2000$.