SOLUTION: Under ideal conditions, a population of rabbits has an exponential growth rate of 11.7% per day. Consider an initial population of 200 rabbits. Find the exponential growth functi

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Question 593293: Under ideal conditions, a population of rabbits has an exponential growth rate of 11.7% per day. Consider an initial population of 200 rabbits. Find the exponential growth function.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The exponential growth function is

N%28t%29+=+N%5B0%5D%2Ae%5E%28kt%29 where N(t) is the population at time t, N%5B0%5D is the initial population and k is the growth constant (some positive number in the interval [0,1])


In this case, the initial population is 200, so N%5B0%5D=200

So the function N%28t%29+=+N%5B0%5D%2Ae%5E%28kt%29 then becomes N%28t%29+=+200%2Ae%5E%28kt%29

The growth rate is 11.7% per day, so when t=1, the population is now 200 + 0.117*200 = 223.4

So this means we have the equation

223.4+=+200%2Ae%5E%28k%29


Let's solve for k


223.4+=+200%2Ae%5E%28k%29

223.4%2F200+=+e%5E%28k%29

1.117+=+e%5E%28k%29

ln%281.117%29+=+ln%28e%5E%28k%29%29

ln%281.117%29+=+k%2Aln%28e%29

ln%281.117%29+=+k%2A%281%29

ln%281.117%29+=+k

0.110646520087064+=+k

k+=+0.110646520087064

So the growth constant is approximately k+=+0.110647


Therefore, the exponential growth function is


N%28t%29+=+200e%5E%280.110647t%29