Chapter 4.8, Problem 24E

(a)

To determine

To describe:

The table given as a function of the number of years since $1990$ .

Explanation of Solution

Given:

The table is

Year	$1990$	$1995$	$1997$	$1998$	$1999$	$2000$	$2001$	$2002$
Subscribers (millions)	$5.3$	$33.8$	$55.3$	$69.2$	$86.0$	$109.5$	$128.4$	$140.8$

Concept used:

Formula:

The equation is cubic

  $y=a{x}^3+b{x}^2+cx+d$

Calculation:

The table is

Year	$1990$	$1995$	$1997$	$1998$	$1999$	$2000$	$2001$	$2002$
Subscribers (millions)	$5.3$	$33.8$	$55.3$	$69.2$	$86.0$	$109.5$	$128.4$	$140.8$

  $y=a{x}^3+b{x}^2+cx+d$

  $\begin{array}{l}a=0.05\\ b=0.6\\ c=2.47\\ d=0.99\end{array}$

  $y=0.05x^3+0.6x^2+2.47x+0.99$

Draw the graph

(b)

To determine

To explain:

The subscribers were in $1996$ .

Answer to Problem 24E

  $y=30.63\text{million}$

Explanation of Solution

Given:

The table is

Year	$1990$	$1995$	$1997$	$1998$	$1999$	$2000$	$2001$	$2002$
Subscribers (millions)	$5.3$	$33.8$	$55.3$	$69.2$	$86.0$	$109.5$	$128.4$	$140.8$

Concept used:

Formula:

The equation is cubic

  $y=a{x}^3+b{x}^2+cx+d$

Calculation:

The table is

Year	$1990$	$1995$	$1997$	$1998$	$1999$	$2000$	$2001$	$2002$
Subscribers (millions)	$5.3$	$33.8$	$55.3$	$69.2$	$86.0$	$109.5$	$128.4$	$140.8$

  $y=a{x}^3+b{x}^2+cx+d$

  $\begin{array}{l}a=0.05\\ b=0.6\\ c=2.47\\ d=0.99\end{array}$

  $y=0.05x^3+0.6x^2+2.47x+0.99$

Putting $x=12$ in above equation

  $\begin{array}{l}y=0.05{\left(12\right)}^3+0.6{\left(12\right)}^2+2.47\left(12\right)+0.99\\ y=30.63\text{million}\end{array}$

Draw the graph

(c)

To determine

To explain:

The $200\text{millions}$ subscribers.

Answer to Problem 24E

  $0=0.05x^3-0.6x^2+2.47x-149.01$

Explanation of Solution

Given:

The table is

Year	$1990$	$1995$	$1997$	$1998$	$1999$	$2000$	$2001$	$2002$
Subscribers (millions)	$5.3$	$33.8$	$55.3$	$69.2$	$86.0$	$109.5$	$128.4$	$140.8$

Concept used:

Formula:

The equation is cubic

  $y=a{x}^3+b{x}^2+cx+d$

Calculation:

The table is

Year	$1990$	$1995$	$1997$	$1998$	$1999$	$2000$	$2001$	$2002$
Subscribers (millions)	$5.3$	$33.8$	$55.3$	$69.2$	$86.0$	$109.5$	$128.4$	$140.8$

  $y=a{x}^3+b{x}^2+cx+d$

  $\begin{array}{l}a=0.05\\ b=0.6\\ c=2.47\\ d=0.99\\ f\left(x\right)=150\end{array}$

  $y=0.05x^3+0.6x^2+2.47x+0.99$

Putting $f\left(x\right)=150$ in above equation

  $\begin{array}{l}150=0.05x^3-0.6x^2+2.47x+0.99\\ 0=0.05x^3-0.6x^2+2.47x-149.01\end{array}$