Question

this exercise uses the exponential growth model.
a grey squirrel population was introduced in a certain county of great britain 35 years ago. biologists observe that the population doubles every 7 years, and now the population is 72,000.
(a) what was the initial size of the squirrel population?
2250 × squirrels
(b) estimate the squirrel population 10 years from now. (round your answer to the nearest whole number.)
× squirrels
(c) of the squirrel population. (assume ( t = 0 ) corresponds to the initial introduction.)

Explanation:

Step1: Determine the exponential growth formula

The general formula for exponential growth is \(n(t)=n_0\cdot a^{\frac{t}{T}}\), where \(n_0\) is the initial population, \(a\) is the growth factor, \(t\) is the time elapsed, and \(T\) is the time it takes for the quantity to change by a factor of \(a\). Here, \(a = 2\) (since the population doubles) and \(T=7\) years.

Step2: Solve for the initial population \(n_0\) for part (a)

We know that currently \(t = 35\) years and \(n(35)=72000\). Substitute into the formula:

$$
LATEXBLOCK0
$$

Step3: Estimate the population 10 years from now for part (b)

Now, the current time \(t = 35\) years. Ten years from now, \(t=35 + 10=45\) years. Using \(n_0 = 2250\), \(a = 2\), \(T = 7\)

$$
LATEXBLOCK1
$$

Answer:

(a) \(2250\) squirrels
(b) \(204815\) squirrels