simple differentials

[joey123]

Let P(t) represent the number of wolves in a population at time t years, when t 0. The population P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k.

a) If P(0)=500, find P(t)

b) If P(2)=700, find k.

c) Find lim P(t) as t goes to infinity. Then expalin what this result represents.

 

[Unco]

Well, have you solved the DE?

Which parts are you having trouble with?

 

[joey123]

I solved it but i am not sure about the answer. My DE was P(t)= -300e^(kt) + 800. What i am having trouble with is part c.

 

[Unco]

Cheers for that.

Your k is missing a negative sign; otherwise good.

Still got to do (b) to get k.

And for (c), (with the corrected P(t)) think of what happens to the exponential as t gets large.

 

[joey123]

Thanks for the help. For part b i got k=.549 roughly based on my calcualtor readings. I thought about what will happen to the equation as t increases, but i dont know exactly how to "find lim P(t) as t goes to infinity.

 

[Unco]

Good stuff.

As t goes to infinity, e^(-0.549 * t) goes to zero, right?

So -300e^(-0.549t) begins being reasonably negative but whimpers out.


Conclusion: P(t) = 800 - 300e^(-0.549t) tend to what as t tends to infinity?

 

[joey123]

800?

 

[Unco]

Brilliant.

 

[joey123]

thanks for all the help