Chapter 4.6, Problem 3E

Population Growth These exercises use the population growth model.

3. Squirrel Population A grey squirrel population was introduced in a certain county of Great Britain 30 years ago. Biologists observe that the population doubles every 6 years, and now the population is 100,000.

1. (a) What was the initial size of the squirrel population?
2. (b) Estimate the squirrel population 10 years from now.
3. (c) Sketch a graph of the squirrel population.

To determine

To find: The initial size of the squirrel population.

Answer to Problem 3E

The initial size of the squirrel population is $3125$.

Explanation of Solution

Given:

The present squirrel population is $100000$.

The squirrel population is double in $6\text{hours}$.

The time $t$ is $30\text{years}$.

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 squirrel population is $100000$, the squirrel population is double in $6\text{hours}$ and the time $t$ is $30\text{years}$.

Substitute $100000$ for $n\left(t\right)$, $30$ for $t$ and $6$ for $a$ in equation (I) to find the number of initial squirrel population,

$\begin{array}{c}100000=n_o\left(2^{\frac{30}{6}}\right)\\ 100000=n_o\left(2^5\right)\\ n_o=\frac{100000}{32}\\ n_o=3125\end{array}$

Thus, the initial size of the squirrel population is $3125$.

To determine

To find: The squirrel population after $10\text{years}$.

Answer to Problem 3E

The squirrel population after $10\text{years}$ from now is $317480$.

Explanation of Solution

Given:

The initial squirrel population is $3125$.

The squirrel population is double in $6\text{hours}$.

Calculation:

The time $t$ is $40$ because squirrel population introduced $30$ years ago and to calculate the squirrel population after $10$ years the total period is $40$ years.

Substitute $3125$ for $n_0$, $40$ for $t$ and $6$ for $a$ in equation (1) to find the squirrel population after $10$ years from now,

$\begin{array}{c}n\left(40\right)=3125\left(2^{\frac{40}{6}}\right)\\ =3125\left(2^{\frac{20}{3}}\right)\\ =3125\left(101.593667\right)\\ =317480.21037\end{array}$

Thus, the squirrel population after $10\text{years}$ from now is $317480$.

To determine

To sketch: The graph of squirrel population.

Explanation of Solution

The squirrel population growth function is $n\left(t\right)=3125\left(2^{\frac{t}{6}}\right)$. The table of values of $n\left(t\right)$ at different values of $t$.

$t$	$0$	$10$	$20$	$30$	$40$	$50$
$n\left(t\right)$	$3125$	$9921.26$	$31498.03$	$100000$	$317480.21$	$107936.84$

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

Thus, the above figure show the graph of $n\left(t\right)=3125\left(2^{\frac{t}{6}}\right)$. The $x\text{-axis}$ denotes the time $t$ and $y\text{-axis}$ denotes $n\left(t\right)$ the squirrel population at time $t$.