1050 CHAPTER 11 Further Topics in Algebra Solve each problem. See Example 3. 95. (Modeling) Insect Population Suppose an insect population density, in thousands per acre, during year n can be modeled by the recursively defined sequence a1 = 8 an = 2.9an-1 - 0.2an-1 2, for n 71. (a) Find the population for n = 1, 2, 3. (b) Graph the sequence for n = 1, 2, 3, c , 20. Use the window 30, 214 by 30, 144. Interpret the graph. 96. Male Bee Ancestors One of the most famous sequences in mathematics is the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, c . (Also see Exercise 33.) Male honeybees hatch from eggs that have not been fertilized, so a male bee has only one parent, a female. On the other hand, female honeybees hatch from fertilized eggs, so a female has two parents, one male and one female. The number of ancestors in consecutive generations of bees follows the Fibonacci sequence. Draw a tree showing the number of ancestors of a male bee in each generation following the description given above. 97. (Modeling) Bacteria Growth If certain bacteria are cultured in a medium with sufficient nutrients, they will double in size and then divide every 40 minutes. Let N1 be the initial number of bacteria cells, N2 the number after 40 minutes, N3 the number after 80 minutes, and Nj the number after 401 j - 12 minutes. (Data from Hoppensteadt, F., and C. Peskin, Mathematics in Medicine and the Life Sciences, SpringerVerlag.) (a) Write Nj+1 in terms of Nj for j Ú 1. (b) Determine the number of bacteria after 2 hr if N1 = 230. (c) Graph the sequence Nj for j = 1, 2, 3, c , 7, where N1 = 230. Use the window 30, 104 by 30, 15,0004. (d) Describe the growth of these bacteria when there are unlimited nutrients. 98. (Modeling) Verhulst’s Model for Bacteria Growth Refer to Exercise 97. If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and growth will slow. According to Verhulst’s model, the number of bacteria Nj at time 401 j - 12 in minutes can be determined by the sequence Nj+1 = c 2 1 + Nj Kd Nj , where K is a constant and j Ú 1. (Data from Hoppensteadt, F., and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.) (a) If N1 = 230 and K = 5000, make a table of Nj for j = 1, 2, 3, c , 20. Round values in the table to the nearest integer. (b) Graph the sequence Nj for j = 1, 2, 3, c , 20. Use the window 30, 204 by 30, 60004. (c) Describe the growth of these bacteria when there are limited nutrients. (d) Make a conjecture about why K is called the saturation constant. Test the conjecture by changing the value of K in the given formula.
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