The document contains tables showing the growth of bacteria populations over time under different scenarios. In each scenario, the bacteria multiply at regular time intervals: - In the first scenario, the bacteria doubles every hour, resulting in populations of 10, 20, 40, 80 at times 0, 1, 2, 3 hours. - In the second, the bacteria increases by a factor of 1.4142135 every half hour, matching the first scenario's results at whole hour intervals. - In the third scenario, the bacteria doubles every quarter hour, again matching the first scenario's whole hour results. The algebraic rules shown confirm that all three scenarios produce identical bacterial population results at whole hour time intervals, demonstrating

1. A bacteria gets *16 every hour
hours
bacteria
0
10
1
2
3

2. A bacteria gets *16 every hour
hours
bacteria
0
10
1
160
2
2560
3
40960

3. A bacteria gets *4 every half-hour
hours
bacteria
0
10
.5
1
1.5
2
2.5
3

4. A bacteria gets *4 every half-hour
hours
bacteria
0
10
.5
40
1
160
1.5
640
2
2560
2.5
10240
3
40960

5. Coincidence?
hours
bacteria
hours
bacteria
0
10
0
10
.5
40
1
160
1.5
640
2
2560
2.5
10240
3
40960
1
2
3
160
2560
40960

6. A bacteria gets doubled every ¼ hour
hours
bacteria
0
10
.25
.5
.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3

7. A bacteria gets doubled every ¼ hour
hours
bacteria
0
10
.25
20
.5
40
.75
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
20480
3
40960

8. Coincidence?
hours
bacteria
hours
bacteria
0
10
0
10
.25
20
.5
40
.75
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
20480
3
40960
.5
1
1.5
2
2.5
3
40
160
640
2560
10240
40960

9. 2 coincidences?!?!?
hours
bacteria
hours
bacteria
hours
bacteria
0
10
0
10
0
10
.25
20
.5
40
.75
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
20480
3
40960
.5
1
160
1
1.5
2
2560
2
2.5
3
40960
3
40
160
640
2560
10240
40960

10. Are these 3 situations identical?
hours
bacteria
hours
bacteria
hours
bacteria
0
10
0
10
0
10
.25
20
.5
40
.75
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
20480
3
40960
.5
1
160
1
1.5
2
2560
2
2.5
3
40960
3
40
160
640
2560
10240
40960

11. Are these 3 situations identical?
hours
bacteria
0
10
hours
bacteria
hours
bacteria
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
160
10
.75
1
0
10
0
According to these .25
tables, yes they are! .5
.5
40
20480
3
40960
1
1.5
2
2560
2
2.5
3
40960
3
160
640
2560
10240
40960
20
40

12. Let’s see what the algebra has to say:
x
0
10
1
160
2
2560
3
40960

13. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
*16 every hour
1
160
2
2560
3
40960

14. Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
0
10
*16 every hour
1
160
2
2560
3
40960

15. Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
x
0
10
0
10
.5
40
= 10(4)1
1
160
= 10(4)2
1.5
640
= 10(4)3
2
2560
= 10(4)4
2.5
10240
= 10(4)5
3
40960
= 10(4)6
1
2
3
160
2560
40960

16. Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
x
y = 10(1.41)2x
0
10
0
10
.5
40
= 10(4)1
quadrupled every
half hour
 After each hour,
it’s happened twice
3
2560
40960
160
= 10(4)2
640
= 10(4)3
2
2560
= 10(4)4
2.5
2
160
1
1.5
1
10240
= 10(4)5
3
40960
= 10(4)6

17. Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
x
y = 10(4)2x
0
10
0
10
.5
40
= 10(4)1
quadrupled every
half hour
 After each hour,
it’s happened twice
3
2560
40960
160
= 10(4)2
640
= 10(4)3
2
2560
= 10(4)4
2.5
2
160
1
1.5
1
10240
= 10(4)5
3
40960
= 10(4)6

18. Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
x
y = 10(4)2x
x
0
10
0
10
0
10
.25
20
.5
40
.75
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
20480
3
40960
.5
1
160
1
1.5
2
2560
2
2.5
3
40960
3
40
160
640
2560
10240
40960

19. doubled every quarter hour
 After each hour, it’s happened 4
times
Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
x
y = 10(4)2x
x
y = 10(1.189)4x
0
10
0
10
0
10
.25
20
.5
40
.75
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
20480
3
40960
.5
1
160
1
1.5
2
2560
2
2.5
3
40960
3
40
160
640
2560
10240
40960

20. doubled every quarter hour
 After each hour, it’s happened 4
times
Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
x
y = 10(4)2x
x
y = 10(2)4x
0
10
0
10
0
10
.25
20
.5
40
.75
80
1
160
1.25
320
1.5
640
1.75
1280
2
2560
2.25
5120
2.5
10240
2.75
20480
3
40960
.5
1
20
1
1.5
2
40
2
2.5
3
80
3
40
160
640
2560
10240
40960

21. A bacteria gets doubled every hour
hours
bacteria
0
10
1
2
3

22. A bacteria gets doubled every hour
hours
bacteria
0
10
1
20
2
40
3
80

23. A bacteria gets
*
1.4142135’d
every half-hour
hours
bacteria
0
10
.5
1
1.5
2
2.5
3

24. A bacteria gets
*
1.4142135’d
every half-hour
hours
bacteria
0
10
.5
14.14
1
20
1.5
28.28
2
40
2.5
56.56
3
80

25. A bacteria gets
*
1.4142135’d
every half-hour
hours
bacteria
hours
bacteria
0
10
0
10
.5
14.14
1
20
1.5
28.28
2
40
2.5
56.56
3
80
1
2
3
20
40
80

26. A bacteria gets
*
1.18920711’d
every ¼ hour
hours
bacteria
0
10
.25
.5
.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3

27. A bacteria gets
*
1.18920711’d
every ¼ hour
hours
bacteria
0
10
.25
11.89
.5
14.14
.75
16.82
1
20
1.25
23.78
1.5
28.28
1.75
33.63
2
40
2.25
47.56
2.5
56.56
2.75
67.26
3
80

28. A bacteria gets
*
1.18920711’d
every ¼ hour
hours
bacteria
hours
bacteria
0
10
0
10
.25
11.89
.5
14.14
.75
16.82
1
20
1.25
23.78
1.5
28.28
1.75
33.63
2
40
2.25
47.56
2.5
56.56
2.75
67.26
3
80
.5
1
1.5
2
2.5
3
14.14
20
28.28
40
56.56
80

29. A bacteria gets
*
1.18920711’d
every ¼ hour
hours
bacteria
hours
bacteria
hours
bacteria
0
10
0
10
0
10
.25
11.89
.5
14.14
.75
16.82
1
20
1.25
23.78
1.5
28.28
1.75
33.63
2
40
2.25
47.56
2.5
56.56
2.75
67.26
3
80
.5
1
20
1
1.5
2
40
2
2.5
3
80
3
14.14
20
28.28
40
56.56
80

30. Are these 3 situations identical?
hours
bacteria
hours
bacteria
hours
bacteria
0
10
0
10
0
10
.25
11.89
.5
14.14
.75
16.82
1
20
1.25
23.78
1.5
28.28
1.75
33.63
2
40
2.25
47.56
2.5
56.56
2.75
67.26
3
80
.5
1
20
1
1.5
2
40
2
2.5
3
80
3
14.14
20
28.28
40
56.56
80

31. Are these 3 situations identical?
hours
bacteria
0
10
hours
bacteria
hours
bacteria
16.82
1
20
1.25
23.78
1.5
28.28
1.75
33.63
2
40
2.25
47.56
2.5
56.56
2.75
20
11.89
.75
1
0
10
0
According to these .25
tables, yes 14.14 are! .5
they
.5
67.26
3
80
1
1.5
2
40
2
2.5
3
80
3
20
28.28
40
56.56
80
10
14.14

32. Let’s see what the algebra has to say:
x
0
10
1
20
2
40
3
80

33. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
Doubled every hour
1
20
2
40
3
80

34. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
Doubled every hour
1
20
2
40
3
80

35. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
x
0
10
0
10
0.5
14.14
= 10(1.4142135)1
1
20
= 10(1.4142135)2
1.5
28.28
= 10(1.4142135)3
2
40
= 10(1.4142135)4
2.5
56.56
= 10(1.4142135)5
3
80
= 10(1.4142135)6
1
2
3
20
40
80

36. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
x
y = 10(1.41)2x
0
10
0
10
14.14
1.4142135’d every
half
= 10(1.4142135)1 hour
1
20
 After each hour,
it’s happened twice
= 10(1.4142135)2
1.5
28.28
= 10(1.4142135)3
2
40
= 10(1.4142135)4
2.5
56.56
= 10(1.4142135)5
3
80
= 10(1.4142135)6
0.5
1
2
3
20
40
80

37. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
x
y = 10(1.41)2x
0
10
0
10
14.14
1.4142135’d every
half
= 10(1.4142135)1 hour
1
20
 After each hour,
it’s happened twice
= 10(1.4142135)2
1.5
28.28
= 10(1.4142135)3
2
40
= 10(1.4142135)4
2.5
56.56
= 10(1.4142135)5
3
80
= 10(1.4142135)6
0.5
1
2
3
20
40
80

38. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
x
y = 10(2½ )2x
0
10
0
10
0.5
1
20
14.14
(2½ )’d every half
= 10(1.4142135)1 hour
 After each hour,
it’s happened twice
= 10(1.4142135)2
3
80
28.28
= 10(1.4142135)3
2
40
= 10(1.4142135)4
2.5
40
20
1.5
2
1
56.56
= 10(1.4142135)5
3
80
= 10(1.4142135)6

39. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
x
y = 10(2½ )2x
x
0
10
0
10
0
10
.25
11.89
.5
14.14
.75
16.82
1
20
1.25
23.78
1.5
28.28
1.75
33.63
2
40
2.25
47.56
2.5
56.56
2.75
67.26
3
80
0.5
1
20
1
1.5
2
40
2
2.5
3
80
3
14.14
20
28.28
40
56.56
80

40. Rule using x = #hrs, y = ji#bacteria
= 10(1.189207115)
1
x
y = 10(2)x
0
10
x
=10 (1.189207115)
=10 (1.189207115) 12ji
3
80
1.25
23.78
28.28
1.75
33.63
2
40
47.56
2.5
40
20
2.25
2
16.82
1.5
4ji
14.14
1
= 10(1.189207115) 3ji
11.89
.75
20
10
.5
1
0
.25
= 10(1.189207115) 2ji
56.56
2.75
67.26
3
80

41. 1.18920711’d every quarter hour
 After each hour, it’s happened 4
times
1ji
= 10(1.189207115)
y = 10(2½ )2x
x
y = 10(1.189)4x
Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
x
0 10(1.189207115) 2ji
10
=
1.5
2
40
28.28
=10 (1.189207115)
2
40
2.5
56.56
=10 (1.189207115) 12ji
3
80
3
80
4ji
1
20
1.25
23.78
1.5
1
20
= 10(1.189207115) 3ji
16.82
28.28
1.75
33.63
2
40
2.25
20
14.14
.75
1
11.89
.5
14.14
10
.25
0.5
0
47.56
2.5
56.56
2.75
67.26
3
80

42. 1.18920711’d every quarter hour
 After each hour, it’s happened 4
times
1ji
= 10(1.189207115)
y = 10(2½ )2x
x
y = 10(1.189)4x
Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
x
0 10(1.189207115) 2ji
10
=
1.5
2
40
28.28
=10 (1.189207115)
2
40
2.5
56.56
=10 (1.189207115) 12ji
3
80
3
80
4ji
1
20
1.25
23.78
1.5
1
20
= 10(1.189207115) 3ji
16.82
28.28
1.75
33.63
2
40
2.25
20
14.14
.75
1
11.89
.5
14.14
10
.25
0.5
0
47.56
2.5
56.56
2.75
67.26
3
80

43. (21/4)’d every quarter hour
 After each hour, it’s happened 4
times
Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
x
y = 10(2½ )2x
x
y = 10(21/4)4x
0
10
0
10
0
10
.25
11.89
.5
14.14
.75
16.82
1
20
1.25
23.78
1.5
28.28
1.75
33.63
2
40
2.25
47.56
2.5
56.56
2.75
67.26
3
80
0.5
1
20
1
1.5
2
40
2
2.5
3
80
3
14.14
20
28.28
40
56.56
80