Question

Rabbits are known for being extremely prolific—having a high reproductive rate. The average gestation period for rabbits is about 28 days and they have an unusual reproductive system that permits a female rabbit to be pregnant with two litters at once. This is an adaptation that allows rabbits to survive in the wild where they have a high mortality rate—they’re a source of food for many species of predators! In situations where there are no natural predators, two adult rabbits could easily become 1000 in one year’s time. i. Using this information, construct an exponential model to predict the number of rabbits after “x” years if there are no natural predators. ii. If the rabbit population were left unchecked (no predators and an abundance of food), how many rabbits would there be at the end of 3 years?

Answer 1

(i) The exponential growth model to predict the number of rabbits after 'x' years is $A = 2e^{6.21460809842x}$.

(ii) The number of rabbits at the end of 3 years will be 250000000.

A continuous exponential growth function is of the form $A = Pe^{rt}$, 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 $A = Pe^{rt}$, 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:

$A = 2e^{rx}$ ... (i)

We have been told that after 1 year, the number of rabbits was 1000.

Thus, substituting A = 1000, and x = 1, we get:

$1000 = 2e^{r*1}\\\Rightarrow 1000 = 2e^r\\\Rightarrow e^r = 1000/2 = 500$

Taking log on both sides, we get:

$log_ee^r = log_e500\\\Rightarrow rlog_ee = log_e500\\\Rightarrow r = 6.21460809842$

Thus, the rate of growth, r = 6.21460809842.

Substituting r = 6.21460809842 in (i), we get:

$A = 2e^{6.21460809842x}$, 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:

$A = 2e^{6.21460809842*3}\\\Rightarrow A = 2e^{18.6438242953}\\\Rightarrow A = 2*125000000\\\Rightarrow A =250000000$

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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