# Population Models

• May 20th 2009, 11:11 PM
function
Population Models
Hey,

I'm having trouble with a population model question. I have been given the differential equation dN/dt=kN; where t is the time elapsed in hours and k is a constant.

I have also been given some initial conditions; when t=0, N=3000 and when t=3, N=9000

With these values i have been asked to find the value of K

I found the equation to be B(t)=3000e^kt, where K= (1/3)ln(3), does this look right? (Worried)

Also i have been given the following:
For t > 3 the culture is washed with a solution that is harmful to the bacteria, and the bacteria are killed of 4500 per hour.

Using this i have been asked to find the number of bacteria in the culture for any time where t>3

I am completely stuck for this part of the question!

Any ideas??

Thanks, Function.
• May 21st 2009, 04:24 AM
tdw
Dude I have the same problem.

I found the equation to be B(t)=3000e^kt, where K= (1/3)ln(3) too

but the second part is confusing.
I tried to solve again by dN/dt= kN - 4500

but failed to solve......
• May 21st 2009, 06:25 PM
Dranalion
Hi function boy,

Did you say you had a problem with changing gears? I might know someone who can help you out

Thanks, Function.
• May 22nd 2009, 07:18 AM
HallsofIvy
Quote:

Originally Posted by tdw
Dude I have the same problem.

I found the equation to be B(t)=3000e^kt, where K= (1/3)ln(3) too

but the second part is confusing.
I tried to solve again by dN/dt= kN - 4500

but failed to solve......

$\displaystyle \frac{dN}{kN- 4500}= dt$
Integrating both sides, $\displaystyle \frac{1}{k}ln(kN- 4500)= t+ C$ so $\displaystyle ln(kN- 4500)= kt+ kC$ and then $\displaystyle kN- 4500= e^{kC}e^{kt}= C'e^{kt}$ where $\displaystyle C'= e^{kC}$.
• May 24th 2009, 12:31 AM
function
Thanks,

I follow what you have done there, but how do i go about solving the variables K and C???? I have tried to use the initial conditions but i dont think that is right, becuase this equation is only for t > 3.

Function
• May 24th 2009, 01:01 AM
tdw
Hey function,
I think the intial condition would be t=3 which is N=9000
coz time starts from t=3 so from t=3 on t= 0
haha.... I dunno whether am I right, that the only way I can think about.
• May 24th 2009, 01:39 AM
function
Hey tdw,
That's what i thought aswell, but that will only give you one variable, and when i tried to do it it just turned into a huge mess (Worried). Did you get an answer in the end?
Function