3. The population P (in thousands) of a colony of rabbits at time (t) years is given by

Question

P(t) = 5 + ae−bt, t ≥ 0
where a and b are positive constants. If it is given that the initial population is 10, 000 rabbits and
one year later this population has reduced by 1, 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 two-thirds of the initial popu￾lation.
(c) The time when there is no change in the rabbit population as time changes. Explain your answer.

Answer 1

since P(1) = 1000
and P(0) = 10000
5+a = 10000
5+9995e^-b = 1000
e^-b = 995/9995 = 0.0995
b = -ln 0.0995 = 2.31
so P(t) = 5 + 9995 e^(-2.31 t)

now you can answer the questions.