Chapter 11.7, Problem 23E

(a)

To determine

To find:

The relationship between in $y$ and $x$ for a power function.

Answer to Problem 23E

  $\ln y=\ln \left(c\right)+a\ln \left(x\right)$

Explanation of Solution

Given:

The function

  $y=x\cdot x^a$

Concept used:

Formula:

The equation is

  $a^x=e^{\left(\ln a\right)x}$

Calculation:

The table is

Altitude (m)	$500$	$1000$	$5000$	$10,000$	$15,000$
Viewing distance(km)	$89$	$126$	$283$	$400$	$490$

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

From equation (1)

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

Therefore, the linear function is

  $\ln y=\ln \left(c\right)+a\ln \left(x\right)$

(b)

To determine

To show:

The table of the distance that a person can see from an airplane depends on the altitude of the airplane.

Answer to Problem 23E

  $N=1.0091\left(e^{\mathrm{.0.0197}}\right)x$.

Explanation of Solution

Given:

The function

  $y=x\cdot x^a$

Concept used:

Formula:

The equation is

  $N=N_0{e}^{kt}$

Calculation:

The table is

Altitude (m)	$500$	$1000$	$5000$	$10,000$	$15,000$
Viewing distance(km)	$89$	$126$	$283$	$400$	$490$

The natural logarithms of both the $x$ and $y$ values

$\ln x$	$6.21$	$6.91$	$8.52$	$9.21$	$9.62$
$\ln y$	$4.49$	$4.48$	$5.65$	$5.99$	$6.19$

(c)

To determine

To find:

The linear regression for the linearized data.

Answer to Problem 23E

  $\ln y=1.3901+0.4994\ln \left(x\right)$

Explanation of Solution

Given:

The function

  $y=x\cdot x^a$

Concept used:

Formula:

The equation is

  $N=N_0{e}^{rt}$

Calculation:

The table is

Altitude (m)	$500$	$1000$	$5000$	$10,000$	$15,000$
Viewing distance(km)	$89$	$126$	$283$	$400$	$490$

  $\begin{array}{l}y=c{x}^a\\ \ln y=\ln {\left(cx\right)}^a\\ \ln y=\ln \left(c\right)+\ln {\left(x\right)}^a\\ \ln y=1.3901+0.4994\ln \left(x\right)\end{array}$

(d)

To determine

To find:

The linear regression for $y$ and $x$ for a power function the linearized data.

Answer to Problem 23E

  $y=4.0153x$

Explanation of Solution

Given:

The function

  $y=x\cdot x^a$

Concept used:

Formula:

The equation is

  $N=N_0{e}^{rt}$

Calculation:

The table is

Altitude (m)	$500$	$1000$	$5000$	$10,000$	$15,000$
Viewing distance(km)	$89$	$126$	$283$	$400$	$490$

  $\begin{array}{l}y=c{x}^a\\ \ln y=\ln {\left(cx\right)}^a\\ \ln y=\ln \left(c\right)+\ln {\left(x\right)}^a\\ \ln y=1.3901+0.4994\ln \left(x\right)\\ e^{\ln y}=e^{1.3901+0.4994\ln \left(x\right)}\\ y=4.0153x\end{array}$