Chapter 11.3, Problem 83E

Solution

a)

To draw: The scatter plot of the data in Table 11.5 shown in the textbook.

For the given table the beginning of months in L1 are entered in the graphing calculator and number of rabbits are entered in L2.

Given information: The given Table 11.5 is shown in the textbook.

Calculation:

For the given table the beginning of months in L1 are entered in the graphing calculator and number of rabbits are entered in L2.

Thus, the scatter plot is plotted below:

  

b)

To find: A logistic regression model for the data and the limit of that model as time approaches infinity.

The logistic regression model for the data is $f\left(x\right)\approx \frac{57.71}{1+6.39e^{-0.19x}}$ and the limit is $\underset{x\to \infty }{\lim }f\left(x\right)\approx 57.71$ .

Given information: The given Table 11.5 is shown in the textbook.

Calculation:

The logistic regression model is found using the logistic feature as $f\left(x\right)\approx \frac{57.71}{1+6.39e^{-0.19x}}$ and as $x$ increases positively large then $6.39e^{-0.19x}$ approaches $0$ so the limit is.

  $\begin{array}{l}\underset{x\to \infty }{\lim }f\left(x\right)\approx \frac{57.71}{1+0}\\ \underset{x\to \infty }{\lim }f\left(x\right)\approx 57.71\end{array}$

Thus, the logistic regression model for the data is $f\left(x\right)\approx \frac{57.71}{1+6.39e^{-0.19x}}$ and the limit is $\underset{x\to \infty }{\lim }f\left(x\right)\approx 57.71$ .

c)

To conclude: The limit of the rabbit population growth in the country.

The limit of the rabbit population growth in the country is about 58,000.

Given information: The given Table 11.5 is shown in the textbook.

Calculation:

Since $f\left(x\right)$ is in thousands so the limit of the rabbit population growth in the country is about 58,000 (rounded to the nearest thousand).

d)

To provide: The reasonable explanation for the population growth limit.

The main reasons for the population growth limit are competition, predation and diseases among the animals.

Given information: The given Table 11.5 is shown in the textbook.

Calculation:

The main reasons for the population growth limit are density dependent factors like competition, predation and diseases among the animals.