Question

Time of a Rabbit population is proportional to square of P.At t=0, there are 12 rabbits and 24 rabbits in 10 years after that. When will there be 48 Rabbits?

Answer 1

Problem 8.1. The time rate of change of a rabbit population P is proportional to the square root of P. At ti 0 the population numbers 100 rabbits and is increasing at the rate of 20 rabbits per month. How many rabbits will be there one year later? Problem 8.2. Suppose that the fish population P(t) in a lake is at- tacked by a disease at time t 0, with the result that the fish cease to reproduce (so that the birth rate is B 0) and the death rate (deaths per week per fish) is thereafter proportional to 1/VP. If there were initially 900 fish in the lake and 441 were left after 6 weeks, how long will it take all the fish in the lake to die? 59 Problem 8.3. The time rate of change of an alligator population P in a swamp is proportional to the square of P. The swamp contained 12 alligators at timet nd 24 alligators 10 years later. How long will it take an alligator population to reach the number 48? Problem 8.4. Suppose that the population P(t) of a country satisfies the logistic differential equation dP kP(200 -P) dt where k is constant. Its population in 1940 was 100 million and was then growing at the rate of 1 million per year. Predict this country's population for the year 2000.

Answer 2

In 15 years there will be 48 rabbits.

Step-by-step explanation:

Given

At any time, the rate of change of rabbit population is proportional to square of population P

Thus,

$\frac{dP}{dt}\propto P^2$

$\implies \frac{dP}{dt}=kP^2$

$\implies \frac{dP}{P^2}=kdt$

$\implies \int_12^PP^{-2}dP=k\int_0^tdt$

$\implies -\frac{1}{P}\Bigr|_12^P=kt$

$\implies -(\frac{1}{P}-\frac{1}{12})=kt$

$\implies kt=(\frac{1}{12}-\frac{1}{P})$

Given, at t = 10, P = 24

Therefore,

$k\times 10=(\frac{1}{12}-\frac{1}{24})$

$\implies 10k=\frac{1}{24}$

$\implies k=\frac{1}{240}$

Now, if P = 48

$\frac{1}{240}t=(\frac{1}{12}-\frac{1}{48})$

$\implies t=240\times \frac{3}{48}$

$\implies t=15$ years

Therefore, in 15 years there will be 48 rabbits.

Hope this answer is helpful.

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