Respuesta :
(i) The exponential growth model to predict the number of rabbits after 'x' years is [tex]A = 2e^{6.21460809842x}[/tex].
(ii) The number of rabbits at the end of 3 years will be 250000000.
A continuous exponential growth function is of the form [tex]A = Pe^{rt}[/tex], where A is the final amount, which was initially P, growing continuously at the rate of r, after a time of t years.
In the question, we are asked to construct an exponential model to predict the number of rabbits after 'x' years.
We assume the exponential growth model to be a continuous model, of the form [tex]A = Pe^{rt}[/tex], where A is the final amount of rabbits, which was initially P, growing continuously at the rate of r, after a time of t years.
The initial quantity of rabbits, P = 2.
Time, t = x years.
Substituting the values, we get:
[tex]A = 2e^{rx}[/tex] ... (i)
We have been told that after 1 year, the number of rabbits was 1000.
Thus, substituting A = 1000, and x = 1, we get:
[tex]1000 = 2e^{r*1}\\\Rightarrow 1000 = 2e^r\\\Rightarrow e^r = 1000/2 = 500[/tex]
Taking log on both sides, we get:
[tex]log_ee^r = log_e500\\\Rightarrow rlog_ee = log_e500\\\Rightarrow r = 6.21460809842[/tex]
Thus, the rate of growth, r = 6.21460809842.
Substituting r = 6.21460809842 in (i), we get:
[tex]A = 2e^{6.21460809842x}[/tex], which is the required model.
To find the number of rabbits at the end of 3 years, we put x = 3, in the above equation to get:
[tex]A = 2e^{6.21460809842*3}\\\Rightarrow A = 2e^{18.6438242953}\\\Rightarrow A = 2*125000000\\\Rightarrow A =250000000[/tex]
Thus, there would be 250000000 rabbits at the end of 3 years.
Learn more about the exponential growth model at
https://brainly.com/question/13223520
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