in a field the population of rabbits grows at a rate proportional to its populat

ID: 2900431 • Letter: I

Question

in a field the population of rabbits grows at a rate proportional to its population. there are originally 150 rabbits in the field and in the absence of outside factors there would be 400 rabbits in one months time. each week predators eat 27 rabbits and 8 rabbits migrate out of the area. after 6 weeks the predators start eating 37 rabbits per week and the same number of rabbits still migrate out of the area. will the rabbits survive to implement their plan for world dominations? if they don't when does the population die out?

Explanation / Answer

Let the Growth rate of rabbit population be given by dG/dw;

Where dG is the growth of rabit in time dw in weeks

initially 150 after 1 month 400 ;

so as per the given condition

dG/dw * w = dP;

where dP is the change in population ;

integrating both sides

LHS limits w =1 to w=4;

RHS limits P=150 to P=400;

integrating we get

G * ln 4 = 400 - 1500;

G = 108.22 in 4 weeks ;

so dG/dW = 27.05 per week;

2. Now predators eat 27 per week

so for the next 6 weeks the population of rabits as per the condition specified will be :

400 + 27.05 *6 - (27*6) - 8*6 = 352.3 = 352 rabbits;

3. predators now eat 37/week

so the Time after which the population dies out = x

352 + 27.05 * x - 37 * x - 8*x =0;

solving for x we get

x= 19.6 weeks

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