Chapter 4, Problem 10P

(a)

To determine

To sketch: The scatter plot of the data point $\left(x,y\right)$.

Explanation of Solution

Given:

$x$	$y$
10	29
20	82
30	151
40	235
50	330
60	430
70	546
80	669
90	797

The scatter plot of the data $\left(x,y\right)$ listed in the table as shown below in Figure 1.

Form Figure 1, it is noticed that the scatter plot of the data $\left(x,y\right)$ listed in the table is an increasing curve.

(b)

To determine

To sketch: The scatter plots of “ $x$ verses $\ln y$” and “ $\ln x$ verses $\ln y$” of the data $\left(x,y\right)$ listed in table.

Explanation of Solution

Given:

The data is listed as follows.

$x$	$y$
10	29
20	82
30	151
40	235
50	330
60	430
70	546
80	669
90	797

Calculation:

The table of $\left(x,\ln y\right)$ and $\left(\ln x,\ln y\right)$ of the data point $\left(x,y\right)$ is listed below.

$x$	$y$	$\ln x$	$\ln y$
10	29	2.3025851	3.3672958
20	82	2.9957323	4.4067192
30	151	3.4011974	5.0172798
40	235	3.6888795	5.4595855
50	330	3.912023	5.7990927
60	430	4.0943446	6.0637852
70	546	4.2484952	6.302619
80	669	4.3820266	6.5057841
90	797	4.4998097	6.6808547

Use online graphing calculator and draw the scatter plot of “ $x$ verses $\ln y$” as shown below in Figure 2.

From Figure 2, it is noticed that the scatter plot of “ $x$ verses $\ln y$” is an increasing function.

In the same manner, use online graphing calculator and draw the scatter plot of “ $\ln x$ verses $\ln y$” as shown below in Figure 3.

From Figure 3, it is noticed that the scatter plot of “ $\ln x$ verses $\ln y$” is an increasing function.

(c)

To determine

To explain: The functions which are more appropriate for modeling of this data, exponential or power function.

Answer to Problem 10P

The power function is more appropriate for modeling of the given data.

Explanation of Solution

The form of the power function is $y=a{x}^b$.

Take the $\ln$ both sides of the power law,

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

Here the natural log form of the power law $y=a{x}^b$ is $\ln y=\ln a+b\ln x$, which represents the straight line in “ $\ln x$ verses $\ln y$” coordinate plan in intercept form.

From the Figure 3, it is noticed that the scatter plot of “ $\ln x$ verses $\ln y$” is a straight line.

Thus, the power function has the best fit of the data point.

(d)

To determine

To find: The appropriate function to model the data.

Answer to Problem 10P

The power function $y=0.89x^{1.5}$ has the best fits of the data.

Explanation of Solution

Given:

The data is listed below in table.

$x$	$y$
10	29
20	82
30	151
40	235
50	330
60	430
70	546
80	669
90	797

Calculation:

Use online graphing calculator and draw the graph of power function $y=a{x}^b$ that give the best fit of the data point shown below in Figure 4.

From Figure 4, it is noticed that the power function $y=0.89x^{1.5}$ (blue line) has the best fit of the data point $\left(x,y\right)$, because of this function passes through all the points of the scatter plot.

Therefore, the power model $y=0.89x^{1.5}$ has best fits of the data.