In light of Dave Slomer's problem of assigning "difficult" problems, I sat down today and worked a practice AP exam from the new Barron's review book before having my students try it. (It's our school vacation week). I ran into a problem on part A of the multiple choice (NO calculators) that I'm not sure how to do without a calculator. I only have a copy of the problem (not the book with the solutions) at home with me, so I'm looking for a little help. Here's the problem:
A population of rabbits grows according to the differential equation dR/dt = 0.001 R (200 - R) where R(t) is the number of rabbits after t months. If there were initially 25 rabbits, aproximately how many months will it take the population to double?
(a) 2.7 (B) 3.5 (C) 4.2 (D) 6.9 (E) 8.4
I tried two different techniques: First, I tried to use Euler's method by hand with one month intervals. I had to do a lot of rounding to get numbers I could work with, and then just guessed based on the answer choices. I saw that the population was initially growing at around 5 rabbits/ month; since I was below half the limiting population, the growth rate was increasing, so it would take a little less than five months, guess choice (C).
Next, I actually solved the diff eq to get: R/(200-R)=(1/7) e^(t/5) Substituting R=50, you get that the doubling time is t = 5 ln(7/3). Again, I estimated; ln (7/3) is a little less than 1, so we guess (C) again.
So, anyone with too much time on their hands, and more insight than I want to suggest an easier way to get the answer for this one.
Thanks in advance for your help.
Elisse Ghitelman Newton North High School Newton, MA, USA