Chapter 4.6, Problem 4E

(a)

To determine

To find: The initial size of the bird population.

Answer to Problem 4E

The initial size of the bird population is $2298$.

Explanation of Solution

Given:

The present bird population is $13000$.

The bird population is double in every $10\text{years}$.

The time $t$ is $25\text{years}$.

Formula used: Exponential growth model (doubling time)

$n\left(t\right)=n_0{2}^{\frac{t}{a}}$ (1)

Where, $n_0$ is the initial size of the population, $a$ is the doubling time, $n\left(t\right)$ is size of the population at time $t$.

Calculation:

The present bird population is $13000$, the bird population is double in every $10\text{years}$ and the time $t$ is $25\text{years}$.

Substitute $13000$ for $n\left(t\right)$, $25$ for $t$ and $10$ for $a$ in equation (I) to find the number of initial bird population,

$\begin{array}{c}13000=n_o\left(2^{\frac{25}{10}}\right)\\ 13000=n_o\left(2^{\frac{5}{2}}\right)\\ n_o=\frac{13000}{5.656854}\\ n_o=2298.09714\end{array}$

Thus, the initial size of the bird population is $2298$.

(b)

To determine

To find: The bird population after $5\text{years}$.

Answer to Problem 4E

The bird population after $5\text{years}$ from now is $18384$.

Explanation of Solution

Given:

The initial bird population is $2298$.

The bird population is double in every $10\text{years}$.

Calculation:

The value of $t$ is $30$ because bird was introduced $25$ years ago and to calculate the bird population after five the total time period is $30$ years.

Substitute $2298$ for $n_0$, $30$ for $t$ and $10$ for $a$ in equation (I) to find the squirrel population after $5$ years from now,

$\begin{array}{c}n\left(30\right)=2298\left(2^{\frac{30}{10}}\right)\\ =2298\left(2^3\right)\\ =2298\left(8\right)\\ =18384\end{array}$

Thus, the bird population after $5\text{years}$ from now is $18384$.

(c)

To determine

To sketch: The graph of bird population.

Explanation of Solution

The bird population growth function is $n\left(t\right)=2298\left(2^{\frac{t}{10}}\right)$. The table of values of $n\left(t\right)$ at different values of $t$.

$t$	$0$	$10$	$20$	$30$	$40$	$50$
$n\left(t\right)$	$2298$	$4596$	$9192$	$18384$	$36768$	$73536$

From the above table the values of $t$ denoted on $x\text{-axis}$ and the value of $n(t)$ denoted on $y\text{-axis}$.

Figure (1)

Thus, the above figure show the graph of $n\left(t\right)=2298\left(2^{\frac{t}{10}}\right)$. The $x\text{-axis}$ denotes the time $t$ and $y\text{-axis}$ denotes $n\left(t\right)$ the bird population at time $t$.