The Fly-by-Night Toxic Waste Disposal Company specializes in the disposal of Agent Yucky. Unfortunately, the company simply dumps Agent Yucky in (formerly) Clear Lake. Fortunately they have been caught. As a result of their activity the current concentration of Agent Yucky in Clear Lake is now 10 ppm (parts per million). Clear Lake is part of a chain of rivers and lakes. Fresh water flows into Clear Lake from upstream and the contaminated water in Clear Lake flows downstream. Scientists working for the Department of Environmental Protection predict that the level of contamination in Clear Lake will fall by 20% each year. Thus, we have the following model for the concentration of Agent Yucky in Clear Lake. This is an example of a discrete dynamical system.
p(1) = 10
p(n + 1) = 0.80 p(n)
This model leads to the following predictions
year pollution year pollution year pollution year pollution
1 10.00 6 3.28 11 1.07 16 0.35
2 8.00 7 2.62 12 0.86 17 0.28
3 6.40 8 2.10 13 0.69 18 0.23
4 5.12 9 1.68 14 0.55 19 0.18
5 4.10 10 1.34 15 0.44 20 0.14
Reproduce the calculations above and draw a graph showing p(1) ... p(20) using your CAS window.
Lawyers for the Fly-by-Night Toxic Waste Disposal Company look at these predictions and argue their client has not done any real damage. They note that Clear Lake will eventually return to its former clear and unpolluted state. They even call in a mathematician who writes the following on a blackboard
Lim p(n) = 0 n --> oo
and explains that this bit of mathematics means, descriptively, that after many years the concentration of Agent Yucky will, indeed, approach zero.
Concerned citizens from a nearby town boo the mathematician's testimony. Fortunately, one of them has taken calculus and knows a little bit about limits. She notes that although "after many years the concentration of Agent Yucky will approach zero" the townspeople like to swim in the lake. State regulations prohibit swimming unless the concentration of Agent Yucky is below 2 ppm. She draws the following graph
and notes that until the concentration of Agent Yucky drops into the green zone the lake will be unsafe for swimming. Looking at the table and the graph she notes that for eight years the lake will be unsafe for swimming and proposes a fine of $100,000 per year for each year that the lake is unsafe -- a total of $800,000.
She questions the mathematician as follows.
Your testimony was correct as far as it went but I remember from studying calculus that talking about the eventual concentration of Agent Yucky after many, many years is only a small part of the story. The more precise meaning of your equation
Lim p(n) = 0 n --> oois that given some tolerance T for the concentration of Agent Yucky there is some time N which may be very far in the future such that for all n >= N
|p(n)| < T
Because we want to swim in the lake we are interested in the tolerance
T = 2 ppm
and we see that
N = 9
which means that in nine years we will be able to return to our old swimming hole. Meanwhile we won't be able to swim in Clear Lake for eight long, hot summers.
Her words were greeted by applause from the audience. The town manager sprang to his feet and noted that although a tolerance of 2 ppm was fine for swimming, the town had been using Clear Lake as a source of drinking water and until the concentration of Agent Yucky dropped below 0.5 ppm its water was unsafe for drinking. He drew the following graph.
and looking at the table
year pollution year pollution year pollution year pollution
1 10.00 6 3.28 11 1.07 16 0.35
2 8.00 7 2.62 12 0.86 17 0.28
3 6.40 8 2.10 13 0.69 18 0.23
4 5.12 9 1.68 14 0.55 19 0.18
5 4.10 10 1.34 15 0.44 20 0.14
noted that it would be safe to drink the water when n >= 15. He proposed a fine of $1,400,000 or $100,000 for each of the 14 years the water was unfit for drinking.
This example points out two aspects of limits and, in particular, limits of the form
Lim p(n) = 0 n --> oo
In practice water, even the original water in Clear Lake, has some pollution. Suppose the natural concentration of Agent Yucky in Clear Lake is 0.1 ppm. In this situation scientists would predict the future concentration of Agent Yucky in the lake to be something like
p(1) = 10 ppm
p(n + 1) = 0.1 + 0.80[p(n) - 0.1]
and any fine would be based not on the amount of pollution in the lake but on the difference between the level of pollution and the natural level 0.1. In other words if the tolerance were T then the company would be fined for each year that
|p(n) - 0.1| >= T
For Problems 7 and 8 determine the how long the company must wait before until the fines cease.