Fish in a Lake

In a certain lake, fishing is prohibited except for during the month of January. Each year, at the beginning of the harvest (January 1) and at the end (January 31) the fish are counted. The last several years have been recorded and are shown on the chart.

	Jan 1st count	# of fish caught	Jan 31st count	growth
Year 1	5000		5000
Year 2	5000		4000
Year 3	6000		4000
Year 4	6000		3000
Year 5	6000		6000
Year 6	3000	(Leave blank.)

The reason for the increases and decreases in the fish population is that there is a limited supply of natural resources in the lake. When some of these resources are being unused, the fish thrive and grow. When the population is larger than what the resources can provide, many fish die.

From a fisherman’s point of view, the more fish—the more food. Therefore, our goal is to figure out how to maximize the number of fish that can be caught each January, while making sure we don’t decrease the next year’s harvest. The following steps will help us accomplish this goal.

1. The percent of the lake’s resources that are being used is directly related to the fish population at that time. When the Jan 31 population is 5000 fish, 100% of the resources are being used, and the result is no growth. Make a chart like the one shown below. Figure out what percent of the lake’s resources are being used or unused for each year. Then figure out the percent increase in the fish population for each year.

2. Since there is a direct relationship between any Jan 31 population and the growth that follows, we should be able to make predictions about the growth based on our previous findings. Show these in a chart like the one shown below.

   	Jan 31st count	% of res. used	% of res. unused	% increase in fish (Feb–Dec)
   Year 1	5000
   Year 2	4000
   Year 3	4000
   Year 4	3000
   Year 5	6000

3. In order to maximize the number of fish caught per year, we must figure out the highest possible population increase in a year. Letting x equal the Jan 31 count, we can express the growth as a function of x. This should say f(x) = and the right part of the equation should show the expression from the bottom-right corner of the chart above. Write this function in standard form. Draw a rough sketch of the graph of this function. Then use this graph to find the maximum value of the function.

4. Given that the lake starts out with a January 1 Year 1 count of 5000, devise a strategy that will allow you to catch the maximum possible number of fish for many years to come. Then duplicate the following chart and fill in the information. According to your plan, how many fish will be caught in the first six years combined? (Hint: It should be more than 18,000 fish.)

   	Jan 1st count	# of fish caught	Jan 31st count	growth
   Year 1	5000
   Year 2
   Year 3
   Year 4
   Year 5
   Year 6

BONUS QUESTION: Just How Do We Count These Fish Anyway?
You want to count the number of fish in a certain lake. You remove and tag 1000 fish and then return them to the lake. A few months later you remove 1000 fish again. You count that 125 of the fish have tags. Then you return these to the lake. Can you make a guess about how many fish live in the lake? (10 points)