Chapter 5.8, Problem 23AYU

(A)

To determine

To find:

The carrying capacity of the environment

Answer to Problem 23AYU

The carrying capacity is the numerator of the logistic function $C=500$

Explanation of Solution

Given:

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

Concept used:

If $x$ and $b$ are positive real numbers and $b\ne 1$

  ${\log }_{b}x=y$ Is same value $b^y=x$

Calculation:

In exponential form using the definition of a logarithm

Using the formula

  ${\log }_{b}x=y$ Is same value $b^y=x$

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

The carrying capacity is the numerator of the logistic function

So, $C=500$

(B)

To determine

To find:

The growth rate of the bald eagle

Answer to Problem 23AYU

The growth rate is the absolute value k thus the growth rate is $16.2\%$

Explanation of Solution

Given:

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

Concept used:

If $x$ and $b$ are positive real numbers and $b\ne 1$

  ${\log }_{b}x=y$ Is same value $b^y=x$

Calculation:

In exponential form using the definition of a logarithm

Using the formula

  ${\log }_{b}x=y$ Is same value $b^y=x$

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

The carrying capacity is the numerator of the logistic function

So, $C=500$

The growth rate is the absolute value k thus the growth rate is $16.2\%$

(C)

To determine

To find:

The carrying capacity of the environment

Answer to Problem 23AYU

The carrying capacity is the numerator of the logistic function $C=500$

Explanation of Solution

Given:

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

Concept used:

If $x$ and $b$ are positive real numbers and $b\ne 1$

  ${\log }_{b}x=y$ Is same value $b^y=x$

Calculation:

In exponential form using the definition of a logarithm

Using the formula

  ${\log }_{b}x=y$ Is same value $b^y=x$

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

The carrying capacity is the numerator of the logistic function

So, $C=500$

(D)

To determine

To find:

The carrying capacity of the environment

Answer to Problem 23AYU

The carrying capacity is the numerator of the logistic function $p\left(3\right)\approx 10eagles$

Explanation of Solution

Given:

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

Concept used:

If $x$ and $b$ are positive real numbers and $b\ne 1$

  ${\log }_{b}x=y$ Is same value $b^y=x$

Calculation:

In exponential form using the definition of a logarithm

Using the formula

  ${\log }_{b}x=y$ Is same value $b^y=x$

  $\begin{array}{l}P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}\\ puttingt=3\\ P\left(3\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}\\ p\left(3\right)\approx 10eagles\end{array}$

(E)

To determine

To find:

The carrying capacity of the environment

Answer to Problem 23AYU

The carrying capacity is the numerator of the logistic function $t\approx 29.7years$

Explanation of Solution

Given:

  $P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}$

Concept used:

If $x$ and $b$ are positive real numbers and $b\ne 1$

  ${\log }_{b}x=y$ Is same value $b^y=x$

Calculation:

In exponential form using the definition of a logarithm

Using the formula

  ${\log }_{b}x=y$ Is same value $b^y=x$

  $\begin{array}{l}P\left(t\right)=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}\\ 300=\frac{1000}{1+32.33e^{-\left(0.439t\right)}}\\ 300\left(1+83.33e^{-\left(0.162t\right)}\right)=1000\\ t\approx 29.7years\end{array}$