Chapter 11.7, Problem 18E

(a)

To determine

To find:

A function that models the data.

Answer to Problem 18E

  $y=-826.4217+520.4168\ln x$

Explanation of Solution

Given:

The data of number of the amount.

Concept used:

Formula:

The equation is

  $y=a\times b^x$

Calculation:

The table is

Year	$1960$	$1965$	$1970$	$1975$	$1980$	$1985$	$1990$	$1995$	$2000$
Degrees $\left(1000s\right)$	$392$	$494$	$792$	$923$	$930$	$980$	$1052$	$1160$	$1238$

  $a^x=e^{\left(\ln a\right)x}\mathrm{....................}\left(1\right)$

From equation (1)

  $\begin{array}{l}y=c{x}^a\\ \ln y=\ln {\left(cx\right)}^a\\ \ln y=\ln \left(c\right)+\ln {\left(x\right)}^a\\ \ln y=\ln \left(c\right)+a\ln \left(x\right)\\ y=-826.4217+520.4168\ln x\end{array}$

Therefore, the linear function is $y=-826.4217+520.4168\ln x$

(b)

To determine

To find:

The equation in terms of $x$ not be the number of years since $1960$ .

Answer to Problem 18E

  $x=0$ And $\ln 0$ is undefined.

Explanation of Solution

Given:

The data of number of the amount.

Concept used:

Formula:

The equation is

  $N=N_0{e}^{kt}$

Calculation:

The table is

Year	$1960$	$1965$	$1970$	$1975$	$1980$	$1985$	$1990$	$1995$	$2000$
Degrees $\left(1000s\right)$	$392$	$494$	$792$	$923$	$930$	$980$	$1052$	$1160$	$1238$

  $\begin{array}{l}y=c{x}^a\\ \ln y=\ln {\left(cx\right)}^a\\ \ln y=\ln \left(c\right)+\ln {\left(x\right)}^a\\ \ln y=\ln \left(c\right)+a\ln \left(x\right)\\ y=-826.4217+520.4168\ln x\end{array}$

The year $1960$ is correspond to $x=0$ and $\ln 0$ is undefined.