Answer:
The function that gives the population at time t is:
[tex]P(t)=24(4^{t})[/tex]
Step-by-step explanation:
In 1859 (t = 0) the population of rabbits in the wilds of Australia was P (0) = 24.
Consider P (t) as the population of rabbits at time t.
It is provided that the population of rabbits grew exponentially.
The formula representing the exponential growth is:
[tex]A=A_{0}a^{t}[/tex]
Here,
A₀ = initial value
a = growth rate
t = time.
The formula representing the exponential growth of rabbits in the wilds of Australia is:
[tex]P(t) =P(0)a^{t}[/tex]
It is provided that P (0) = 24.
[tex]P(t)=P(0)a^{t}=24a^{t}[/tex]
It is provided that every 6 months the population of rabbits double.
Then after half a year the population is 48.
Compute the value of a as follows:
[tex]P(\frac{1}{2})=24\times a^{\frac{1}{2}}\\48=24\times a^{\frac{1}{2}}\\2=a^{\frac{1}{2}}\\a=2^{2}\\a=4[/tex]
The function that gives the population at time t is:
[tex]P(t)=24(4^{t})[/tex]
