Chapter 2.5, Problem 78AYU

(a)

To determine

To find: $R\left(0,\right),R\left(3\right),$ and $R\left(5\right)$ , and interpret the values.

Explanation of Solution

Given information:

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.

Calculation:

In order to find $R\left(0\right)$ , replace $x$ with 0 in the given function and simplify.

  $\begin{array}{c}R\left(0\right)=170.7{\left(0\right)}^2+1373\left(0\right)+1080\\ =1080\end{array}$

Find $R\left(3\right)$ by replacing $x$ with 3 in the given function and then simplify.

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

Find $R\left(5\right)$ by replacing $x$ with 5 in the given function and then simplify.

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

We note that $x$ is the number of years after 2005.

Thus, we can say that $R\left(0\right)$ or $1080 represents the total projected world wide digital music revenue for the year 2005.

Similarly, we can say that $R\left(3\right)$ represents the revenue for the year 2008, and $R\left(5\right)$ represents the revenue for the year 2010.

(b)

To determine

To find: the value of $r=R\left(x-5\right)$ .

Answer to Problem 78AYU

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

Explanation of Solution

Given information:

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.

Calculation:

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

We get $R\left(x-5\right)$ as

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

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

(c)

To determine

To find: $r\left(5,\right),r\left(8\right),$ and $r\left(10\right)$ and interpret the values.

Answer to Problem 78AYU

  $r\left(5\right)=1080$ , $r\left(8\right)=6735.3$ and $r\left(10\right)=12212.5$

Explanation of Solution

Given information:

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.

Calculation:

In order to find $r\left(5\right)$ , replace $x$ with 5 in the equation for $r$ and simplify.

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

Find $r\left(8\right)$ by replacing $x$ with 8 in the equation for $r$ and then simplify.

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

Find $r\left(10\right)$ by replacing $x$ with 10 in the equation for $r$ and then simplify.

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

We know that the function $r$ represents the revenue 5 years back when compared to the function $R$ .

For the function $R$ , $x=5$ represents the year 2010. This means that for the function $r$ , $x=5$ represents 2005.

Thus, we can say that $r\left(5\right)$ represents the total projected worldwide digital music revenue for the year 2005.

Similarly, we can say that $r\left(3\right)$ represents the revenue for the year 2008, and $r\left(10\right)$ represents the revenue for the year 2005.

(d)

To determine

To find: the representation for $x$ .

Answer to Problem 78AYU

The number of the years after 2000

Explanation of Solution

Given information:

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.

Calculation:

We nota that in the model $r,x$ represents the number of the years after 2000

(e)

To determine

To find: the advantage in using the model $r$ when estimating the projected revenues for a given year instead of the model $R$ .

Answer to Problem 78AYU

The model $r$ is used we can determine the revenue from the year 2000.

Explanation of Solution

Given information:

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.

Calculation:

The model $R$ is used to determine the total projected worldwide 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.