SOLUTION: Hi. I'm not sure how to solve this problem. I'm assuming it has to do with derivatives. Thank you. On a certain island there are two populations of deer. After t years the numbe

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Hi. I'm not sure how to solve this problem. I'm assuming it has to do with derivatives. Thank you. On a certain island there are two populations of deer. After t years the numbe      Log On


   

Question 721098: Hi. I'm not sure how to solve this problem. I'm assuming it has to do with derivatives. Thank you.
On a certain island there are two populations of deer. After t years the numbers of deer in the two populations are p(t) = 100 e^t and q(t) = 1000 e^-t. When is the total population smallest?
t=________
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The total population is f%28t%29=100e%5Et%2B1000+e%5E%28-t%29
The derivative of that function would be the population growth rate
g%28t%29=100e%5Et-1000+e%5E%28-t%29
When that growth is zero the population will have reached its minimum:
100e%5Et-1000+e%5E%28-t%29=0
If we define x=e%5Et then e%5E%28-t%29=1%2Fx and we can solve
100x-1000%2Fx=0 as a first step
100x-1000%2Fx=0 --> %28100x%5E2-1000%29%2Fx=0 --> 100x%5E2-1000=0 --> x%5E2-10=0 --> x%5E2=10 --> x=sqrt%2810%29
In terms of t
e%5Et=sqrt%2810%29 --> t=ln%28sqrt%2810%29%29 --> t=ln%2810%5E0.5%29 --> t=0.5ln%2810%29 or t=ln%2810%29%2F2= approx 1.151
So the total deer population will reach its minimum after about 1.151 years.