Ex: Exponential Growth Function - Population — Transcript

Full transcript

1. 0:00a growing city had a population of
2. 0:02500,000 in 2005 in 2010 the population
3. 0:07was
4. 0:08760,000 we want to assume exponential
5. 0:11growth and then Express the population
6. 0:13after T years as a function of T predict
7. 0:17the population in
8. 0:192025 then determine in which year the
9. 0:21population will reach 1
10. 0:23million so we want to start by finding
11. 0:25the exponential function that's going to
12. 0:27model this population so we'll be using
13. 0:29this exponential function here where P
14. 0:32of T would be the population after a
15. 0:34certain number of years P Sub 0 or P not
16. 0:37would be the initial
17. 0:39population K would be the exponential
18. 0:41growth rate and T would be the time and
19. 0:44years so let's start by finding the
20. 0:46exponential function for this population
21. 0:49so when we read these first two
22. 0:51sentences the starting population would
23. 0:53be 500,000 so P sub Z or P not is equal
24. 0:56to 500,000
25. 1:01this is in 2005 then in 2010 the
26. 1:05population was 760,000 so P of
27. 1:09T would be equal to
28. 1:13760,000 and then from 2005 to 2010
29. 1:17that's a span of five years so T is
30. 1:20going to be equal to five so now we'll
31. 1:22perform substitution into our
32. 1:24exponential function and then solve for
33. 1:26K our exponential growth rate so we want
34. 1:29to solve solve the equation
35. 1:32760,000 =
36. 1:35500,000 * e ra the power of K * T but
37. 1:40since T is 5 we'll have
38. 1:435K now we want to solve this exponential
39. 1:46equation for K so we'll first isolate
40. 1:48the exponential part so we'll divide
41. 1:50both sides by
42. 1:52500,000 this simplifies to
43. 1:55one and then on the left side of 760,000
44. 2:02ID
45. 2:05500,000 which is equal to
46. 2:111.52 so we have 1.52 would equal e to
47. 2:15the power of
48. 2:175K and now since we have base e here
49. 2:20instead of taking the common log of both
50. 2:22sides we'll take the natural log of both
51. 2:25sides and on the right side we can apply
52. 2:28the power property of log to move this
53. 2:305K to the
54. 2:32front so now we'd have natural log
55. 2:371.52
56. 2:39equals 5K * natural log e but natural
57. 2:44log e is equal to 1 if we have natural
58. 2:47log e this is log base e and since e
59. 2:52raised to the first power is equal to
60. 2:55e this is equal to
61. 2:58one so this simplifies to one so to
62. 3:01solve this for K we just need to divide
63. 3:03both sides by
64. 3:05five of course we could divide both
65. 3:07sides by natural log e but again that's
66. 3:10just equal to one so it's not going to
67. 3:11change
68. 3:13anything so we have K is equal to this
69. 3:16quotient which we'll have to get a
70. 3:18decimal approximation
71. 3:21for so we have natural log of
72. 3:261.52 / 5
73. 3:31and the more decimal places that we use
74. 3:32for K the more accurate our answer is
75. 3:34going to be let's go ahead and take this
76. 3:36out to six decimal
77. 3:38places so we'll have
78. 3:410.083
79. 3:45742 and now we have our exponential
80. 3:47function that's going to model this
81. 3:48population we'll have P of T is equal to
82. 3:52the initial population of
83. 3:555,000 * e ra the power of 0 8374298849
84. 4:29P of 20 to approximate the population in
85. 4:33the year
86. 4:472025 so now we'll go back to the
87. 4:51calculator and approximate this
88. 4:56value second natural log brings up e the
89. 5:00power of 8374298849
90. 5:29I so we want to substitute 1 million for
91. 5:32p of T and solve for T so we'll have 1
92. 5:38million equals
93. 5:42500,000 e raised the power of 0.0
94. 5:468374
95. 5:482T so we're going to isolate the
96. 5:50exponential part so we'll divide both
97. 5:52sides by
98. 5:54500,000 simplifies to one this would be
99. 5:572 so we have 2 equals
100. 6:00e raised to this
101. 6:02exponent just as we did before we'll now
102. 6:04take the natural log of both sides of
103. 6:06the equation and then apply the power
104. 6:09property of logarithms so we'll move
105. 6:10this exponent to the front of the
106. 6:16logarithm so we'll have natural log 2
107. 6:19equals
108. 6:210.083 472 T times natural log e but as
109. 6:26we showed before natural log e
110. 6:28simplifies to 1 so to solve this for T
111. 6:32we just need to divide by this decimal
112. 6:37coefficient again this simplifies to one
113. 6:41so let's go back to the calculator this
114. 6:43quotient would be the value of T which
115. 6:45will be the number of years after 2005
116. 7:01so I can see that t is approximately
117. 7:048.28
118. 7:15years so this would be 8.28 years after
119. 7:19the Year
120. 7:202005 and since the base year is the Year
121. 7:232005 2005 plus
122. 7:268.28 means that the population would
123. 7:28reach 1 million
124. 7:30during the
125. 7:31[Music]
126. 7:33year
127. 7:392013 okay I hope you found this helpful