Solve the problem.
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 40(j – 1) in minutes can be determined by the sequence
where K is a constant and j ≥ 1.
(a) If N1 = 230 and K = 5000, make a table of Nj for j = 1, 2, 3, . . . , 20. Round values in the table to the nearest integer.
(b) Graph the sequence Nj for j = 1, 2, 3, . . . , 20. Use the window [0, 20] by [0, 6000].
(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.
(a) Write Nj+1 in terms of Nj for j ≥ 1.
(b) Determine the number of bacteria after 2 hr if N1 = 230.
(c) Graph the sequence Nj for j = 1, 2, 3, . . . , 7, where N1 = 230. Use the window [0, 10] by [0, 15,000].
(d) Describe the growth of these bacteria when there are unlimited nutrients.