Question

The population P (in thousands) of a colony of rabbits at time (t) years is given by P(t) = 3+ae−bt, t ≥0 where a and b are positive constants. If it is given that the initial population is 8,000 rabbits and one year later this population has reduced by 2,000, find the following; (a) The values of a and b. (Do not approximate your answer). (b) The time (to 2 decimal places) when the population becomes half of the initial population. (c) The time when there is no change in the rabbit population as time changes. Explain your answer. er​

Answer 1

Answer:

The population of the day is not a gai 5

Answer 2

Given  : The population P (in thousands) of a colony of rabbits at time (t) years is given by  $P(t)=3+ae^{-bt}$   t ≥0  

a and b are positive constants

Initial population is 8,000 rabbits and one year later this population has reduced by 2,000

To Find :  The values of a and b.

The time (to 2 decimal places) when the population becomes half of the initial population

The time when there is no change in the rabbit population as time changes

Solution:

$P(t)=3+ae^{-bt}$   t ≥0  

Initial population is 8000

=> P(0) = 8  as  P (in thousands) and t = 0 for initial

=> 8 = 3 +  a

=> a = 5

$P(t)=3+5e^{-bt}$  

one year later this population has reduced by 2,000

t = 1

and P(1)  = 8 - 2 = 6

$6=3+5e^{-b}$  

$3=5e^{-b}$  

$\frac{3}{5} =e^{-b}$  

$\frac{5}{3} =e^{b}$  

b = ln (5/3)  ≈ 0.5108

P(t)  = 3  + 5  (3/5)^t

P(t) = 4  ( becomes half)

4 = 3  +  5  (3/5)^t

=> 1/5 = (3/5)^t

=> t ≈ 3.15 sec

Population will approach to 3000 but never touch 3000  and hence there is no such time when there is no change in the rabbit population as time changes

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