Question

The population of a colony of rabbits grows exponentially. The colony begins with 5 rabbits; 5 years later there are 310 rabbits.

(a) Express the population of the colony of rabbits, P , as a function of time, t , in years.

(b) Use the graph to estimate how long it takes for the population of rabbits to reach 1000 rabbits.

Answer 1

P = A e^kt
A = 5 is given when t = 0 (because e^0 = 1)
P = 5 e^kt
310 = 5 e^5k
62 = e^5k
ln 62 = 5 k
5 k = 4.127
k = .8254
so
P = 5 e^.8254 t

now if P = 1,000
200 = e^.8254 t
ln 200 = 5.298 = .8254 t
t = 6.42 years