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?
x squirrels
(b) estimate the squirrel population 10 years from now. (round your answer to the nearest whole number.)
x squirrels
(c) sketch a graph of the squirrel population. (assume ( t = 0 ) corresponds to the initial introduction.)

Explanation:

Step1: Determine the exponential growth formula

The exponential growth model is \(n(t)=n_0\cdot2^{\frac{t}{k}}\), where \(n(t)\) is the population at time \(t\), \(n_0\) is the initial population, and \(k\) is the doubling - time. Given \(k = 7\) years.

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

We know that now \(t = 35\) years and \(n(35)=72000\). Substitute into the formula \(n(t)=n_0\cdot2^{\frac{t}{k}}\):

$$72000=n_0\cdot2^{\frac{35}{7}}$$

$$72000=n_0\cdot2^{5}$$

$$72000=n_0\cdot32$$

$$n_0=\frac{72000}{32}=2250$$

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

Now \(t = 35\) years, 10 years from now \(t=35 + 10=45\) years. Using \(n_0 = 2250\) and \(k = 7\) in the formula \(n(t)=n_0\cdot2^{\frac{t}{k}}\):

$$n(45)=2250\cdot2^{\frac{45}{7}}$$

$$n(45)=2250\cdot2^{6.4286}$$

$$n(45)=2250\times91.02$$

$$n(45)\approx204800$$

Answer:

(a) \(2250\) squirrels
(b) \(204800\) squirrels