solve population growth

Question

Under ideal conditions, a population of rabbits has an exponential growth rate of 11.7% per day. Consider an initial population of 100 rabbits.
(a) Find the exponential growth function
(b) What will the population be after one week? After 2 weeks?
(c) Find the doubling time.

Answer 1

P = P₀e^kt

 

In this case

 

(a) P = (100)e^(0.117)(t)

 

(b) P = (100)e^{(0.117)(7 days)}

 

     P = 226.8

 

(c) 200 = (100)e^(0.117)t

 

     e^(0.117)(t) = 2

 

      (0.117)(t) = ln(2)

 

     t = 5.92 days

 

 

Answer 2

Start by noting the variables:

A(t)=A*e^kt

Where k is the growth rate (in decimals) and t is the time (in days)  A is the original amount or initial population, and A(t) represents the population at time t.

So,

The growth function is:

A(t)=100e^.117t

 

The population at time t=1 week (7 days) and t=2 weeks (14 days)

is:

For 1 week, t=7

A(7)=100e^.117*7=226.8 or 226

For 2 weeks, t=14

A(14)=100e.117*14=514.4 or 514

Finally the doubling time is:

 

200=100e^.117*t

So 2=e^.117t

take the ln of both sides:

ln(2)=.117t

t=ln(2)/.117= 5.92 days, or 6 days.