Chapter 3.3, Problem 87E

To determine

To find: the logarithmic equation that relates an animal’s weight x (in pounds) and its lowest stride frequency while galloping y (in strides per minute).

Answer to Problem 87E

  $\ln y=-0.14\ln x+5.7$

Explanation of Solution

Given:

The table:

x	25	35	50	75	500	1000
y	191.5	182.5	173.8	164.2	125.9	114.2

Concept Used:

Slope of a line:

The slope of line passing through two point $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$ , $x_1\ne x_2$

Slope intercept form of equation of line is $y=mx+b$ , where m is the slope of the line and $\left(0,b\right)$ is the y -intercept of the line.

Construct a new table by taking the natural logarithms of the x and y values in the given table.

$\ln x$	3.219	3.555	3.912	4.317	6.215	6.908
$\ln y$	5.255	5.208	5.158	5.101	4.835	4.738

Now plot each point $\left(\ln x,\ln y\right)$ to make sure they lie on a line.

  

Now, choose any two points to determine the slope of the line.

So, using the points $\left(3.219,5.255\right)$ and $\left(3.555,5.208\right)$ , the slope of the line is:

  $\begin{array}{c}m=\frac{y{2}_{}-y_1}{x_2-x_1}\\ =\frac{5.208-5.255}{3.555-3.219}\\ =-\frac{0.047}{0.336}\\ \approx -0.14\end{array}$

Now, from the slope-intercept form of line, the equation of line is:

  $Y=mX+b$

Where $Y=\ln y$ and $x=\ln x$

That is, equation of line is,

  $\ln y=0.14\ln x+b$

Now, to find b , substitute any point $\left(x,y\right)$ from the original table, so let’s substitute $\left(25,191.5\right)$ ,

  $\begin{array}{c}\ln \left(191.5\right)=-0.14\ln \left(25\right)+b\\ 5.25=-0.14\left(3.22\right)+b\\ b=5.25+0.14\left(3.22\right)\\ b\approx 5.7\end{array}$

Thus, the equation of line is:

  $\ln y=-0.14\ln x+5.7$

Conclusion:

The logarithmic equation that relates y to x is $\ln y=-0.14\ln x+5.7$ .