A wildlife preservation group records the number of squirrels in a city park. The first count recorded is 30 squirrels. A second count of 60 squirrels is recorded six months later, and 120 squirrels are counted at the end of one year. Assuming that no squirrel dies and the growth rate remains the same, which function represents the number of squirrels after x years?

From the statement we know that the squirrel's population doubles every six months, it started with 30 squirrels, after six months there are 60 squirrels and six months later (end of one year) there is a population of 120 squirrels.

So, doubles every six months (rate) , and 6 months are 0.5 year

Writing the equation:

[tex]SquirrelsPopulation=StartPopulation*(rate)^{\frac{x}{0.5}}[/tex]

Where,

[tex]StartPopulation=30\\rate=2\\x=time(years)[/tex]

Then, substituting the rate into the equation, we have:

[tex]SquirrelsPopulation=StartPopulation*(2)^{\frac{x}{0.5}}[/tex]

Let's prove that the equation works

Evaluating the population after: 0.5 year, 1 year, 1.5 years and 2 years:

Population after 0.5 year:

[tex]SquirrelsPopulation=30*(2)^{\frac{0.5}{0.5}}=30*2^{1}=60[/tex]

Population after 1 year:

[tex]SquirrelsPopulation=30*(2)^{\frac{1}{0.5}}=30*2^{2}=120[/tex]

Population after 1.5 years:

[tex]SquirrelsPopulation=30*(2)^{\frac{1.5}{0.5}}=30*2^{3}=240[/tex]

Population after 2 years:

[tex]SquirrelsPopulation=30*(2)^{\frac{2}{0.5}}=30*2^{4}=480[/tex]

So, the equation works.

Have a nice day!

astha8579 astha8579 · Answer 2 · 2022-02-14T08:06:12+01:00

You can use the fact that squirrels will have exponential growth with base 2.

The function which represents the number of squirrels after t years is

[tex]P(t) = 30.(4)^t[/tex]

How to determine population growth function?

Suppose that we have

[tex]r[/tex] = rate of population growth
[tex]P_i[/tex] = initial population size
[tex]t[/tex] = time elapsed
[tex]f[/tex] = period in which population grows

Then the population growth function is given as

[tex]P(r,t,f) = P_i(r)^{t/f}[/tex]

Since we're given that, for squirrels case given,

[tex]P_i = 30[/tex]

[tex]f = 0.5 \: year[/tex]

[tex]r = 2 \: \: \text{(As population is doubling)}[/tex]

Thus,

After t time, the population is given as

[tex]P(t) = 30(2)^{t/0.5} = 30.(2)^{2t} = 30.(4)^t[/tex] (its just function of t (in years) now as other values are specified and constant)

Thus,

The function which represents the number of squirrels after t years is

[tex]P(t) = 30.(4)^t[/tex]

Learn more about exponential function here:

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