help: derivatives question?

Question

the rate of natural growth of a town is proportional to the population. if the population grows from 20,000 to 50,000 in 10 years, then to the nearest 10th year, when will the population reach 100,000?

but i havent done differntial equations yet, or integration. the answer is 17.6 years.

ok, i've done derivatives, and this falls under "application of derivatives", or the chapter "rates of change"

yeah you were right, there is a formula. thanks a lot, you really helped a lot.

Answer 1

This is more of a differential equation question. Let P be the population, then you know

dP/dt = kP

Integrate it by separating the variables:

dP/P = k dt

ln P = kt + C

p = exp(kt + C) = exp(kt) exp(C) = K exp(kt)

If you look at the equation, it's clear that K is the population at t = 0. Let that be 20,000, then you also know that the population at t = 10 is 50000. That will let you find k, the growth constant. From there, you can set P to 100,000 and solve for t. At this point, it's really an algebra question.

Edit: OK. It would help, then, to know what you have done just recently that led to this question.

Edit 2: Hmm. Well, one thing occurs to me. One is that there may be a section in your book relating to exponential growth, and you'll need to use whatever techniques they give you there. Truthfully, though, I don't hold out much hope for that one - they'd pretty much have to give you a formula to work with. You might check for that.

I can see a way to estimate it using the doubling time, but you'll never get an answer accurate to .1 years that way.

Try this: you know, as I said before, that dP/dt = kP. If they talk about exponential growth at all, they may tell you that problems like this have a solution on the order of

P = K exp(kt)

In other words, they give you the solution to the differential equation I mentioned above. In that case, though, this is just an algebra II problem, with the solution as I outlined above. Using that technique, you get 17.56 years which (of course) rounds to 17.6 years.

Sorry, luv - without that, I don't see any way to the answer with the tools you have.

Answer 2

Its not a derivatives question. The answer is 30 years. You need to figure out the difference between 20000 and 50000 and then divide that by ten to get teh rate per year. Then divide 100000 by the rate you get and round to the nearest ten years