A population grows at a rate proportional to the diﬀerence between its size and a maximum possible of 10 million: $$\large \frac{dP}{dt}=k(10-P)$$ where P is the population measured in millions. In 1960 the population was 2.5 million; in 2000 it was 7.5 million. (a) Find a relation between the population and the time. (b) When will the population reach 9.9 million? (c) What will be the population in 2050? I have tackled questions like this before, but I seem to making a reoccurring mistake .

Question

A population grows at a rate proportional to the diﬀerence between its size and a maximum possible of 10 million:
$$\large \frac{dP}{dt}=k(10-P)$$
where P is the population measured in millions. In 1960 the population was 2.5 million; in 2000 it was 7.5 million.
(a) Find a relation between the population and the time.
(b) When will the population reach 9.9 million?
(c) What will be the population in 2050?

I have tackled questions like this before, but I seem to making a reoccurring mistake >.<

anonymous · Answer

ah looks like a differential equation

ray10 · Answer

Yes that is correct
I think I worked out the first bit but I'm not sure :S

anonymous · Answer

there is actually a subsection for those, but it's ok, i'll try to help but i am not sure if i have time to solve it completely right now.

ray10 · Answer

I arrive at
$$\large p=10-e^{2.0149-0.0275t}$$

anonymous · Answer

hmm i'm not very good at these yet either so i need some time to solve it lol. i can't say if that is correct or not right now.

ray10 · Answer

oh of course, sure thing, take your time :)

anonymous · Answer

you solved it by using an integrating factor?

ray10 · Answer

no not really, just integrated both sides after seperating the variables and solving using the conditions given

anonymous · Answer

well i'm sorry but i really have no clue on this one, i though i would be able to solve it but i really don't know. Either i'm having a bad moment or it's different from the ones i'm used to.

ray10 · Answer

But you stopped by and tried to help me out, and I'm very grateful for that :)

ray10 · Answer

it's different for me too, I'm not use to this one, or my answer is wrong haha

anonymous · Answer

but if you assume it is correct, from that formula b and c shouldn't be hard right?

ray10 · Answer

that is correct :) as long as A is correct, then B and C shall be easy :)

anonymous · Answer

however i think it's a bit weird

anonymous · Answer

you function increases with a decelerating rate, while population growth often increase with an increasing rate.

ray10 · Answer

that is true, that is what I am having trouble understanding :/

anonymous · Answer

i hate not being able to solve a problem lol.

anonymous · Answer

you may still be right though because this looks more like the one for temperature if you look carefully. that one does increase in a decelerating rate.

" A population grows at a rate proportional to the diﬀerence between its size and a maximum possible of 10 million"

anonymous · Answer

so maybe you're right after all, but i really don't know for sure.

ray10 · Answer

I hate not being able to solve it either!
Yeah I seemed to realize there was something wrong, you helped clarify what  it was!
I need to sleep for now, will have to look at that tomorrow

anonymous · Answer

@Ray10 i plotted it on my calculator, along with a function that looks like yours, and also looks simular to the plot, but it still has some flaws. maybe it helps: http://img849.imageshack.us/img849/7851/lxn1.jpg

anonymous · Answer

the red dotted line is the plot of the differential equation, the blue line and the formula are the formula i tried to match the red line with. which only worked partially.

anonymous · Answer

the black lines is a direction field.