Chapter 1, Problem 51RE

(a)

To determine

To find: The value of $p(0)$ and give a possible interpretation of this number.

Answer to Problem 51RE

The value of $p(0)=25$

This could be the number of bears that were originally present at the reserve.

Explanation of Solution

Given information: The population equation $p(t)=\frac{200}{1+7e^{-0.1r}}$

Calculation:

The given population equation $p(t)=\frac{200}{1+7e^{-0.1r}}$

Let $t=0$ in the equation:

  $\begin{array}{c}p(0)=\frac{200}{1+7e^{-0.1(0)}}\\ =\frac{200}{1+7}\\ =25\end{array}$

Hence, the value of $p(0)=25$

Therefore, this could be the number of bears that were originally present at the reserve.

(b)

To determine

To find: The limit $\underset{t\to \infty }{\lim }p(t)$

Answer to Problem 51RE

The value of $\underset{t\to \infty }{\lim }p(t)=200$

Explanation of Solution

Given information: The limit is $\underset{t\to \infty }{\lim }p(t)$

Calculation:

  $t\to \infty ,e^{-0.1t}$ will become smaller and smaller and approach the asymptote $y=0$ .

Thus, calculate the limit as,

  $\begin{array}{c}\underset{t\to \infty }{\lim }p(t)=\frac{200}{1+7\cdot 0}\\ =\frac{200}{1}\\ =200\end{array}$

Therefore, the required limit is $\underset{t\to \infty }{\lim }p(t)=200$ .

(c)

To determine

To find: Give a possible interpretation of the result in part (b)

Answer to Problem 51RE

When the population reaches 200, food supplies begin to run out, preventing future population growth.

Explanation of Solution

Given information: From part (b) $\underset{t\to \infty }{\lim }p(t)=200$

Calculation:

Given that $p(t)$ is a logistic growth function, $p$ approaches the asymptote of $y=200$ as t increases. One explanation is that the population of bears will at first increase to numbers that are gradually approaching 200, but never reach that level. This may be the reserve's capacity to support a bear population indefinitely. When the population reaches 200, food supplies begin to run out, preventing future population growth.