Things have finally quieted down around Guilder since Prince Humperdinck was kicked out of Florin. The good news is that I've managed to find a government job as the head of the First National Park of Guilder. The bad news is that most of the Park consists of the Fire Swamp. The worst news is that the only other employee at the Park is Fezzik. Don't get me wrong, I love the giant like a brother, but he can be a little frustrating to work with. When I went looking for help with our long range planning, your enterprising and resourceful professor naturally referred me to you. We have two species that have me really worried about the future of the Park: the indigenous ROUS (rodents of unusual size) and the BTS (Brown Tree Snakes) which entered Guilder about 50 years ago as a stowaway from Guam on the ship of the Dread Pirate Roberts. Fortunately, ROUS's eat BTS's. Unfortunately, BTS's reproduce very rapidly. My predecessor at the Park was a meticulous census taker (who used statistical sampling, by the way, to get more accurate counts), so I have approximate populations for each species for the last 30 years. Year BTS's ROUS's 1970 15300 415 1972 9890 910 1974 2860 950 1976 3340 525 1978 9340 250 1980 12290 460 1982 9050 830 1984 4840 855 1986 5130 545 1988 8720 340 1990 10490 500 1992 8550 770 1994 6030 790 1996 6200 560 1998 8350 410 2000 9410 525

It looks like the populations are following some sort of pattern, but I'm not sure what it is. My real problem is that when either population gets very large, I will need additional employees to make sure that both species stay within the park and don't escape in the Guilder farmland. This is where I need your expert help. Specifically, I need a prediction for what the populations will be in each of the next 20 years. As a side note, after the Dread Pirate Roberts killed the extremely intelligent (and a bit conceded) Vizzini by poisoning him in a dual of wits, I found the mathematical equations given below on his person. Since Vizzini was formerly the accountant for the First National Park of Guilder, I am wondering if they have anything to do with the ROUS and BTS populations.

R (t) = 981.386(0.96082) ^t *0.5sin(pie(t-1))/ (5)) +550

B (t) =14875.254 * (0.95396) ^t *1/2 cos(pie(t))/ (5) + 7800

I believe the populations are fluctuating less and less and may eventually stabilize. I would like your expert opinion on whether or not the populations do stabilize, and if they do, I need to know how long it will take and what the eventual populations will be. Once the populations stop fluctuating so drastically, we will be able to dramatically improve access to the park by offering summer camps, establishing permanent camp grounds, and perhaps even adding a log ride, although there are still some flame- retardant issues to be worked out. This should all be possible when the ROUS population is fluctuating by less than 75 per year and the BTS population is fluctuating by less than 500 per year. As usual, I need your expert recommendation on when this will occur. Finally, I would appreciate it if you would determine if there seemed to be any point in time(s) when the rate of population growth is zero for both species. I have a meeting with the Budget Advisory Committee at the end of the month to propose our budget for the next two decades, so I would greatly appreciate your report by October 18th. Gratefully yours, Inigo Montoya P.S. A Few Helpful Hints From Your Professor To see the general trend of the populations, I would suggest plotting the points for each population separately, with time on the horizontal axis and population on the vertical axis. It may make things a little bit easier if you let time t=0 be 1970. Be sure to use the tools and proper notation of Calculus in your reply (i.e., limits, rate of change, derivatives, tables, graphs, etc.)