Exponential
functions
A quantity that varies exponentially satisfies two requirements:
1. The increase (or decrease)
of the quantity in each time interval is a constant fraction of the value of
the quantity at the start of that interval.
2. Every increase (or decrease)
is added to (or subtracted from) the value at the start of period and the
result is used to compute the change in the next time period.
For example, a bank account,
in which you make no withdrawals or deposits (after the initial one), is an example
of an exponential function. The interest earned in each time interval is a
constant fraction of the principal; the interest is added to the principal and
the result is used to compute the interest in the next time period. Some credit
cards use the same sort of method – the interest on the unpaid balance is added
to what you owe, and the interest in the next time period is computed on this
sum.
A number of other quantities
behave at least approximately exponentially, including the demand for energy and
the population of a country. No real-world process is exactly exponential since
there are usually many factors that influence the growth rate. However, an
exponential model is a very useful first approximation to predict the future
value of a quantity given its current magnitude and its historical growth rate.
A quantity that is exactly
exponential will double in magnitude after a time approximately given by 70/r,
where r is the growth rate in percent. Thus, an exponential function growing at
a rate of 10% per year will double in size about every 70/10= 7 years; if the
growth rate is 5% per year, the doubling time is 70/5= 14 years. This doubling is a very important feature of
an exponential function, since it produces changes in the quantity that are much
larger than you would estimate based on its initial value. This is because the
doubling is with respect to the current value, not the original value. Thus a
quantity whose size is 1 unit initially will double to 2 units after one
doubling time, 4 units after the next doubling time, 8 units after the next
one, etc. No real-world function can be truly exponential for all time, since
exponential growth will eventually exceed any finite value, and all real-world
functions are finite.
Although an exact calculation
of the behavior of an exponential function requires some calculus and the use
of the natural logarithms, you can get a feel for the behavior of the function
using simple spreadsheets. The following tables are computed using Excel spread
sheets. Each sheet is constructed in the same way; the only difference is the
annual growth rate that I used to compute the values. In constructing the
spreadsheets, I assumed that we have an initial quantity of something that will
last for 600 years at the current rate of consumption. That is, the initial
value for the quantity is 600 units, and the initial rate of consumption is 1
unit per year. If the rate remained constant, then the quantity would last 600
years. (This is the approximate situation with the known reserves of coal.).
The first table assumes that
the rate of consumption will grow by 1% per year. The first column shows the
year number. The second shows the rate of consumption during that year. Each
entry in this column is 1% more than the entry above it. The third column gives
the quantity at the start of the year and the 4th column gives the
quantity remaining at the end of the year. The 5th column gives the
total that was used since the start of the calculation. Thus in the first year
the initial quantity is 600, the rate is 1 and there are 599 units left at the
end of the year. The rate has increased by 1% to 1.01 in the second year, and
so only 597.99 units remain at the end of the year. A total of 2.01 units of
the material were used in the first 2 years. The table continues in this way
until the resource is reduced to 0. The resource is completely exhausted after
195 years. I have also added an “x” at the end of each line where the rate
doubles. This happens in years 71 and 141, which agrees with our expectation
that the rate doubles after 70/1= 70 years.
The next tables are constructed in the same way for annual
growth rates of 2%, 3%, 4% and 5%. Although they are constructed using simple
Excel spreadsheets, they conform very closely to the behavior of a mathematical
exponential function. The final table is based on the one for 3% annual growth,
but it assumes that we “discover” 600 additional units of material in the 48th
year – an increment equal to the total that we had at the start. The extra 600
units extend the life of the resource from 100 years to 122 years. This is not
surprising, since near the end of the 3% table, the annual usage is about 18
units per year, so that the 600 units would extend the lifetime by only about
30+ years even if the rate did not continue to increase.
I have also plotted the rate of consumption and the amount
remaining for 1%, 3% and 5%. These plots are here: Usage per
year, Quantity remaining. Note that there is
not much difference for about the first 25 years, so that it is hard to
estimate the actual rate from the data at the start of the calculation.
Initial= 600.00 Annual Growth Rate= 0.01
Year Rate start end Total used
1 1.00 600.00 599.00 1.00
2 1.01 599.00 597.99 2.01
3 1.02 597.99 596.97 3.03
4 1.03 596.97 595.94 4.06
5 1.04 595.94 594.90 5.10
6 1.05 594.90 593.85 6.15
7 1.06 593.85 592.79 7.21
8 1.07 592.79 591.71 8.29
9 1.08 591.71 590.63 9.37
10 1.09 590.63 589.54 10.46
11 1.10 589.54 588.43 11.57
12 1.12 588.43 587.32 12.68
13 1.13 587.32 586.19 13.81
14 1.14 586.19 585.05 14.95
15 1.15 585.05 583.90 16.10
16 1.16 583.90 582.74 17.26
17 1.17 582.74 581.57 18.43
18 1.18 581.57 580.39 19.61
19 1.20 580.39 579.19 20.81
20 1.21 579.19 577.98 22.02
21 1.22 577.98 576.76 23.24
22 1.23 576.76 575.53 24.47
23 1.24 575.53 574.28 25.72
24 1.26 574.28 573.03 26.97
25 1.27 573.03 571.76 28.24
26 1.28 571.76 570.47 29.53
27 1.30 570.47 569.18 30.82
28 1.31 569.18 567.87 32.13
29 1.32 567.87 566.55 33.45
30 1.33 566.55 565.22 34.78
31 1.35 565.22 563.87 36.13
32 1.36 563.87 562.51 37.49
33 1.37 562.51 561.13 38.87
34 1.39 561.13 559.74 40.26
35 1.40 559.74 558.34 41.66
36 1.42 558.34 556.92 43.08
37 1.43 556.92 555.49 44.51
38 1.45 555.49 554.05 45.95
39 1.46 554.05 552.59 47.41
40 1.47 552.59 551.11 48.89
41 1.49 551.11 549.62 50.38
42 1.50 549.62 548.12 51.88
43 1.52 548.12 546.60 53.40
44 1.53 546.60 545.07 54.93
45 1.55 545.07 543.52 56.48
46 1.56 543.52 541.95 58.05
47 1.58 541.95 540.37 59.63
48 1.60 540.37 538.78 61.22
49 1.61 538.78 537.17 62.83
50 1.63 537.17 535.54 64.46
51 1.64 535.54 533.89 66.11
52 1.66 533.89 532.23 67.77
53 1.68 532.23 530.55 69.45
54 1.69 530.55 528.86 71.14
55 1.71 528.86 527.15 72.85
56 1.73 527.15 525.42 74.58
57 1.75 525.42 523.67 76.33
58 1.76 523.67 521.91 78.09
59 1.78 521.91 520.13 79.87
60 1.80 520.13 518.33 81.67
61 1.82 518.33 516.51 83.49
62 1.83 516.51 514.68 85.32
63 1.85 514.68 512.83 87.17
64 1.87 512.83 510.95 89.05
65 1.89 510.95 509.06 90.94
66 1.91 509.06 507.15 92.85
67 1.93 507.15 505.23 94.77
68 1.95 505.23 503.28 96.72
69 1.97 503.28 501.31 98.69
70 1.99 501.31 499.32 100.68
71 2.01 499.32 497.32 102.68 x
72 2.03 497.32 495.29 104.71
73 2.05 495.29 493.24 106.76
74 2.07 493.24 491.18 108.82
75 2.09 491.18 489.09 110.91
76 2.11 489.09 486.98 113.02
77 2.13 486.98 484.85 115.15
78 2.15 484.85 482.70 117.30
79 2.17 482.70 480.52 119.48
80 2.19 480.52 478.33 121.67
81 2.22 478.33 476.11 123.89
82 2.24 476.11 473.87 126.13
83 2.26 473.87 471.61 128.39
84 2.28 471.61 469.33 130.67
85 2.31 469.33 467.02 132.98
86 2.33 467.02 464.69 135.31
87 2.35 464.69 462.34 137.66
88 2.38 462.34 459.96 140.04
89 2.40 459.96 457.56 142.44
90 2.42 457.56 455.14 144.86
91 2.45 455.14 452.69 147.31
92 2.47 452.69 450.21 149.79
93 2.50 450.21 447.72 152.28
94 2.52 447.72 445.19 154.81
95 2.55 445.19 442.65 157.35
96 2.57 442.65 440.07 159.93
97 2.60 440.07 437.47 162.53
98 2.63 437.47 434.85 165.15
99 2.65 434.85 432.20 167.80
100 2.68 432.20 429.52 170.48
101 2.70 429.52 426.81 173.19
102 2.73 426.81 424.08 175.92
103 2.76 424.08 421.32 178.68
104 2.79 421.32 418.54 181.46
105 2.81 418.54 415.72 184.28
106 2.84 415.72 412.88 187.12
107 2.87 412.88 410.01 189.99
108 2.90 410.01 407.11 192.89
109 2.93 407.11 404.18 195.82
110 2.96 404.18 401.22 198.78
111 2.99 401.22 398.23 201.77
112 3.02 398.23 395.21 204.79
113 3.05 395.21 392.17 207.83
114 3.08 392.17 389.09 210.91
115 3.11 389.09 385.98 214.02
116 3.14 385.98 382.84 217.16
117 3.17 382.84 379.67 220.33
118 3.20 379.67 376.46 223.54
119 3.24 376.46 373.23 226.77
120 3.27 373.23 369.96 230.04
121 3.30 369.96 366.66 233.34
122 3.33 366.66 363.33 236.67
123 3.37 363.33 359.96 240.04
124 3.40 359.96 356.56 243.44
125 3.43 356.56 353.13 246.87
126 3.47 353.13 349.66 250.34
127 3.50 349.66 346.15 253.85
128 3.54 346.15 342.62 257.38
129 3.57 342.62 339.04 260.96
130 3.61 339.04 335.43 264.57
131 3.65 335.43 331.79 268.21
132 3.68 331.79 328.10 271.90
133 3.72 328.10 324.39 275.61
134 3.76 324.39 320.63 279.37
135 3.79 320.63 316.84 283.16
136 3.83 316.84 313.00 287.00
137 3.87 313.00 309.13 290.87
138 3.91 309.13 305.23 294.77
139 3.95 305.23 301.28 298.72
140 3.99 301.28 297.29 302.71
141 4.03 297.29 293.26 306.74 x
142 4.07 293.26 289.20 310.80
143 4.11 289.20 285.09 314.91
144 4.15 285.09 280.94 319.06
145 4.19 280.94 276.75 323.25
146 4.23 276.75 272.52 327.48
147 4.27 272.52 268.24 331.76
148 4.32 268.24 263.92 336.08
149 4.36 263.92 259.56 340.44
150 4.40 259.56 255.16 344.84
151 4.45 255.16 250.71 349.29
152 4.49 250.71 246.22 353.78
153 4.54 246.22 241.68 358.32
154 4.58 241.68 237.10 362.90
155 4.63 237.10 232.47 367.53
156 4.68 232.47 227.79 372.21
157 4.72 227.79 223.07 376.93
158 4.77 223.07 218.30 381.70
159 4.82 218.30 213.48 386.52
160 4.87 213.48 208.62 391.38
161 4.91 208.62 203.70 396.30
162 4.96 203.70 198.74 401.26
163 5.01 198.74 193.73 406.27
164 5.06 193.73 188.67 411.33
165 5.11 188.67 183.55 416.45
166 5.16 183.55 178.39 421.61
167 5.22 178.39 173.17 426.83
168 5.27 173.17 167.90 432.10
169 5.32 167.90 162.58 437.42
170 5.37 162.58 157.21 442.79
171 5.43 157.21 151.78 448.22
172 5.48 151.78 146.30 453.70
173 5.54 146.30 140.76 459.24
174 5.59 140.76 135.17 464.83
175 5.65 135.17 129.52 470.48
176 5.70 129.52 123.82 476.18
177 5.76 123.82 118.05 481.95
178 5.82 118.05 112.23 487.77
179 5.88 112.23 106.36 493.64
180 5.94 106.36 100.42 499.58
181 6.00 100.42 94.42 505.58
182 6.06 94.42 88.37 511.63
183 6.12 88.37 82.25 517.75
184 6.18 82.25 76.07 523.93
185 6.24 76.07 69.84 530.16
186 6.30 69.84 63.53 536.47
187 6.36 63.53 57.17 542.83
188 6.43 57.17 50.74 549.26
189 6.49 50.74 44.25 555.75
190 6.56 44.25 37.69 562.31
191 6.62 37.69 31.07 568.93
192 6.69 31.07 24.38 575.62
193 6.76 24.38 17.62 582.38
194 6.82 17.62 10.80 589.20
195 6.89 10.80 3.91 596.09
Initial= 600.00 Annual Growth Rate= 0.02
Year Rate start end Total used
1 1.00 600.00 599.00 1.00
2 1.02 599.00 597.98 2.02
3 1.04 597.98 596.94 3.06
4 1.06 596.94 595.88 4.12
5 1.08 595.88 594.80 5.20
6 1.10 594.80 593.69 6.31
7 1.13 593.69 592.57 7.43
8 1.15 592.57 591.42 8.58
9 1.17 591.42 590.25 9.75
10 1.20 590.25 589.05 10.95
11 1.22 589.05 587.83 12.17
12 1.24 587.83 586.59 13.41
13 1.27 586.59 585.32 14.68
14 1.29 585.32 584.03 15.97
15 1.32 584.03 582.71 17.29
16 1.35 582.71 581.36 18.64
17 1.37 581.36 579.99 20.01
18 1.40 579.99 578.59 21.41
19 1.43 578.59 577.16 22.84
20 1.46 577.16 575.70 24.30
21 1.49 575.70 574.22 25.78
22 1.52 574.22 572.70 27.30
23 1.55 572.70 571.16 28.84
24 1.58 571.16 569.58 30.42
25 1.61 569.58 567.97 32.03
26 1.64 567.97 566.33 33.67
27 1.67 566.33 564.66 35.34
28 1.71 564.66 562.95 37.05
29 1.74 562.95 561.21 38.79
30 1.78 561.21 559.43 40.57
31 1.81 559.43 557.62 42.38
32 1.85 557.62 555.77 44.23
33 1.88 555.77 553.89 46.11
34 1.92 553.89 551.97 48.03
35 1.96 551.97 550.01 49.99
36 2.00 550.01 548.01 51.99 x
37 2.04 548.01 545.97 54.03
38 2.08 545.97 543.89 56.11
39 2.12 543.89 541.76 58.24
40 2.16 541.76 539.60 60.40
41 2.21 539.60 537.39 62.61
42 2.25 537.39 535.14 64.86
43 2.30 535.14 532.84 67.16
44 2.34 532.84 530.50 69.50
45 2.39 530.50 528.11 71.89
46 2.44 528.11 525.67 74.33
47 2.49 525.67 523.18 76.82
48 2.54 523.18 520.65 79.35
49 2.59 520.65 518.06 81.94
50 2.64 518.06 515.42 84.58
51 2.69 515.42 512.73 87.27
52 2.75 512.73 509.98 90.02
53 2.80 509.98 507.18 92.82
54 2.86 507.18 504.33 95.67
55 2.91 504.33 501.41 98.59
56 2.97 501.41 498.44 101.56
57 3.03 498.44 495.41 104.59
58 3.09 495.41 492.32 107.68
59 3.15 492.32 489.17 110.83
60 3.22 489.17 485.95 114.05
61 3.28 485.95 482.67 117.33
62 3.35 482.67 479.32 120.68
63 3.41 479.32 475.91 124.09
64 3.48 475.91 472.43 127.57
65 3.55 472.43 468.87 131.13
66 3.62 468.87 465.25 134.75
67 3.69 465.25 461.56 138.44
68 3.77 461.56 457.79 142.21
69 3.84 457.79 453.94 146.06
70 3.92 453.94 450.02 149.98
71 4.00 450.02 446.02 153.98 x
72 4.08 446.02 441.94 158.06
73 4.16 441.94 437.78 162.22
74 4.24 437.78 433.54 166.46
75 4.33 433.54 429.21 170.79
76 4.42 429.21 424.79 175.21
77 4.50 424.79 420.29 179.71
78 4.59 420.29 415.69 184.31
79 4.69 415.69 411.01 188.99
80 4.78 411.01 406.23 193.77
81 4.88 406.23 401.35 198.65
82 4.97 401.35 396.38 203.62
83 5.07 396.38 391.31 208.69
84 5.17 391.31 386.13 213.87
85 5.28 386.13 380.86 219.14
86 5.38 380.86 375.47 224.53
87 5.49 375.47 369.98 230.02
88 5.60 369.98 364.38 235.62
89 5.71 364.38 358.67 241.33
90 5.83 358.67 352.84 247.16
91 5.94 352.84 346.90 253.10
92 6.06 346.90 340.84 259.16
93 6.18 340.84 334.65 265.35
94 6.31 334.65 328.35 271.65
95 6.43 328.35 321.92 278.08
96 6.56 321.92 315.35 284.65
97 6.69 315.35 308.66 291.34
98 6.83 308.66 301.83 298.17
99 6.96 301.83 294.87 305.13
100 7.10 294.87 287.77 312.23
101 7.24 287.77 280.52 319.48
102 7.39 280.52 273.13 326.87
103 7.54 273.13 265.60 334.40
104 7.69 265.60 257.91 342.09
105 7.84 257.91 250.07 349.93
106 8.00 250.07 242.07 357.93 x
107 8.16 242.07 233.91 366.09
108 8.32 233.91 225.59 374.41
109 8.49 225.59 217.10 382.90
110 8.66 217.10 208.44 391.56
111 8.83 208.44 199.61 400.39
112 9.01 199.61 190.60 409.40
113 9.19 190.60 181.41 418.59
114 9.37 181.41 172.04 427.96
115 9.56 172.04 162.48 437.52
116 9.75 162.48 152.73 447.27
117 9.95 152.73 142.79 457.21
118 10.14 142.79 132.64 467.36
119 10.35 132.64 122.30 477.70
120 10.55 122.30 111.74 488.26
121 10.77 111.74 100.98 499.02
122 10.98 100.98 90.00 510.00
123 11.20 90.00 78.80 521.20
124 11.42 78.80 67.37 532.63
125 11.65 67.37 55.72 544.28
126 11.89 55.72 43.83 556.17
127 12.12 43.83 31.71 568.29
128 12.37 31.71 19.34 580.66
129 12.61 19.34 6.73 593.27
Initial= 600.00 Annual Growth Rate= 0.03
Year Rate Start end Total used
1 1.00 600.00 599.00 1.00
2 1.03 599.00 597.97 2.03
3 1.06 597.97 596.91 3.09
4 1.09 596.91 595.82 4.18
5 1.13 595.82 594.69 5.31
6 1.16 594.69 593.53 6.47
7 1.19 593.53 592.34 7.66
8 1.23 592.34 591.11 8.89
9 1.27 591.11 589.84 10.16
10 1.30 589.84 588.54 11.46
11 1.34 588.54 587.19 12.81
12 1.38 587.19 585.81 14.19
13 1.43 585.81 584.38 15.62
14 1.47 584.38 582.91 17.09
15 1.51 582.91 581.40 18.60
16 1.56 581.40 579.84 20.16
17 1.60 579.84 578.24 21.76
18 1.65 578.24 576.59 23.41
19 1.70 576.59 574.88 25.12
20 1.75 574.88 573.13 26.87
21 1.81 573.13 571.32 28.68
22 1.86 571.32 569.46 30.54
23 1.92 569.46 567.55 32.45
24 1.97 567.55 565.57 34.43 x
25 2.03 565.57 563.54 36.46
26 2.09 563.54 561.45 38.55
27 2.16 561.45 559.29 40.71
28 2.22 559.29 557.07 42.93
29 2.29 557.07 554.78 45.22
30 2.36 554.78 552.42 47.58
31 2.43 552.42 550.00 50.00
32 2.50 550.00 547.50 52.50
33 2.58 547.50 544.92 55.08
34 2.65 544.92 542.27 57.73
35 2.73 542.27 539.54 60.46
36 2.81 539.54 536.72 63.28
37 2.90 536.72 533.83 66.17
38 2.99 533.83 530.84 69.16
39 3.07 530.84 527.77 72.23
40 3.17 527.77 524.60 75.40
41 3.26 524.60 521.34 78.66
42 3.36 521.34 517.98 82.02
43 3.46 517.98 514.52 85.48
44 3.56 514.52 510.95 89.05
45 3.67 510.95 507.28 92.72
46 3.78 507.28 503.50 96.50
47 3.90 503.50 499.60 100.40
48 4.01 499.60 495.59 104.41 x
49 4.13 495.59 491.46 108.54
50 4.26 491.46 487.20 112.80
51 4.38 487.20 482.82 117.18
52 4.52 482.82 478.30 121.70
53 4.65 478.30 473.65 126.35
54 4.79 473.65 468.86 131.14
55 4.93 468.86 463.93 136.07
56 5.08 463.93 458.85 141.15
57 5.23 458.85 453.61 146.39
58 5.39 453.61 448.22 151.78
59 5.55 448.22 442.67 157.33
60 5.72 442.67 436.95 163.05
61 5.89 436.95 431.05 168.95
62 6.07 431.05 424.99 175.01
63 6.25 424.99 418.74 181.26
64 6.44 418.74 412.30 187.70
65 6.63 412.30 405.67 194.33
66 6.83 405.67 398.84 201.16
67 7.03 398.84 391.80 208.20
68 7.25 391.80 384.56 215.44
69 7.46 384.56 377.09 222.91
70 7.69 377.09 369.41 230.59
71 7.92 369.41 361.49 238.51 x
72 8.16 361.49 353.33 246.67
73 8.40 353.33 344.93 255.07
74 8.65 344.93 336.28 263.72
75 8.91 336.28 327.37 272.63
76 9.18 327.37 318.19 281.81
77 9.45 318.19 308.74 291.26
78 9.74 308.74 299.00 301.00
79 10.03 299.00 288.97 311.03
80 10.33 288.97 278.64 321.36
81 10.64 278.64 268.00 332.00
82 10.96 268.00 257.04 342.96
83 11.29 257.04 245.75 354.25
84 11.63 245.75 234.12 365.88
85 11.98 234.12 222.14 377.86
86 12.34 222.14 209.81 390.19
87 12.71 209.81 197.10 402.90
88 13.09 197.10 184.01 415.99
89 13.48 184.01 170.54 429.46
90 13.88 170.54 156.65 443.35
91 14.30 156.65 142.35 457.65
92 14.73 142.35 127.62 472.38
93 15.17 127.62 112.45 487.55
94 15.63 112.45 96.82 503.18
95 16.10 96.82 80.73 519.27 x
96 16.58 80.73 64.15 535.85
97 17.08 64.15 47.07 552.93
98 17.59 47.07 29.49 570.51
99 18.12 29.49 11.37 588.63
100 18.66 11.37 -7.29 607.29
Initial= 600.00 Annual Growth Rate= 0.04
Year Rate start end Total used
1 1.00 600.00 599.00 1.00
2 1.04 599.00 597.96 2.04
3 1.08 597.96 596.88 3.12
4 1.12 596.88 595.75 4.25
5 1.17 595.75 594.58 5.42
6 1.22 594.58 593.37 6.63
7 1.27 593.37 592.10 7.90
8 1.32 592.10 590.79 9.21
9 1.37 590.79 589.42 10.58
10 1.42 589.42 587.99 12.01
11 1.48 587.99 586.51 13.49
12 1.54 586.51 584.97 15.03
13 1.60 584.97 583.37 16.63
14 1.67 583.37 581.71 18.29
15 1.73 581.71 579.98 20.02
16 1.80 579.98 578.18 21.82
17 1.87 578.18 576.30 23.70
18 1.95 576.30 574.35 25.65
19 2.03 574.35 572.33 27.67 x
20 2.11 572.33 570.22 29.78
21 2.19 570.22 568.03 31.97
22 2.28 568.03 565.75 34.25
23 2.37 565.75 563.38 36.62
24 2.46 563.38 560.92 39.08
25 2.56 560.92 558.35 41.65
26 2.67 558.35 555.69 44.31
27 2.77 555.69 552.92 47.08
28 2.88 552.92 550.03 49.97
29 3.00 550.03 547.03 52.97
30 3.12 547.03 543.92 56.08
31 3.24 543.92 540.67 59.33
32 3.37 540.67 537.30 62.70
33 3.51 537.30 533.79 66.21
34 3.65 533.79 530.14 69.86
35 3.79 530.14 526.35 73.65
36 3.95 526.35 522.40 77.60
37 4.10 522.40 518.30 81.70 x
38 4.27 518.30 514.03 85.97
39 4.44 514.03 509.59 90.41
40 4.62 509.59 504.97 95.03
41 4.80 504.97 500.17 99.83
42 4.99 500.17 495.18 104.82
43 5.19 495.18 489.99 110.01
44 5.40 489.99 484.59 115.41
45 5.62 484.59 478.97 121.03
46 5.84 478.97 473.13 126.87
47 6.07 473.13 467.05 132.95
48 6.32 467.05 460.74 139.26
49 6.57 460.74 454.17 145.83
50 6.83 454.17 447.33 152.67
51 7.11 447.33 440.23 159.77
52 7.39 440.23 432.84 167.16
53 7.69 432.84 425.15 174.85
54 7.99 425.15 417.15 182.85
55 8.31 417.15 408.84 191.16 x
56 8.65 408.84 400.19 199.81
57 8.99 400.19 391.20 208.80
58 9.35 391.20 381.85 218.15
59 9.73 381.85 372.12 227.88
60 10.12 372.12 362.01 237.99
61 10.52 362.01 351.49 248.51
62 10.94 351.49 340.55 259.45
63 11.38 340.55 329.17 270.83
64 11.83 329.17 317.34 282.66
65 12.31 317.34 305.03 294.97
66 12.80 305.03 292.23 307.77
67 13.31 292.23 278.92 321.08
68 13.84 278.92 265.08 334.92
69 14.40 265.08 250.68 349.32
70 14.97 250.68 235.71 364.29
71 15.57 235.71 220.14 379.86
72 16.19 220.14 203.94 396.06 x
73 16.84 203.94 187.10 412.90
74 17.52 187.10 169.59 430.41
75 18.22 169.59 151.37 448.63
76 18.95 151.37 132.42 467.58
77 19.70 132.42 112.72 487.28
78 20.49 112.72 92.23 507.77
79 21.31 92.23 70.92 529.08
80 22.16 70.92 48.76 551.24
81 23.05 48.76 25.71 574.29
82 23.97 25.71 1.73 598.27
Initial= 600.00 Annual Growth Rate= 0.05
Year Rate Start End Total used
1 1.00 600.00 599.00 1.00
2 1.05 599.00 597.95 2.05
3 1.10 597.95 596.85 3.15
4 1.16 596.85 595.69 4.31
5 1.22 595.69 594.47 5.53
6 1.28 594.47 593.20 6.80
7 1.34 593.20 591.86 8.14
8 1.41 591.86 590.45 9.55
9 1.48 590.45 588.97 11.03
10 1.55 588.97 587.42 12.58
11 1.63 587.42 585.79 14.21
12 1.71 585.79 584.08 15.92
13 1.80 584.08 582.29 17.71
14 1.89 582.29 580.40 19.60
15 1.98 580.40 578.42 21.58 x
16 2.08 578.42 576.34 23.66
17 2.18 576.34 574.16 25.84
18 2.29 574.16 571.87 28.13
19 2.41 571.87 569.46 30.54
20 2.53 569.46 566.93 33.07
21 2.65 566.93 564.28 35.72
22 2.79 564.28 561.49 38.51
23 2.93 561.49 558.57 41.43
24 3.07 558.57 555.50 44.50
25 3.23 555.50 552.27 47.73
26 3.39 552.27 548.89 51.11
27 3.56 548.89 545.33 54.67
28 3.73 545.33 541.60 58.40
29 3.92 541.60 537.68 62.32 x
30 4.12 537.68 533.56 66.44
31 4.32 533.56 529.24 70.76
32 4.54 529.24 524.70 75.30
33 4.76 524.70 519.94 80.06
34 5.00 519.94 514.93 85.07
35 5.25 514.93 509.68 90.32
36 5.52 509.68 504.16 95.84
37 5.79 504.16 498.37 101.63
38 6.08 498.37 492.29 107.71
39 6.39 492.29 485.90 114.10
40 6.70 485.90 479.20 120.80
41 7.04 479.20 472.16 127.84
42 7.39 472.16 464.77 135.23
43 7.76 464.77 457.01 142.99
44 8.15 457.01 448.86 151.14 x
45 8.56 448.86 440.30 159.70
46 8.99 440.30 431.31 168.69
47 9.43 431.31 421.88 178.12
48 9.91 421.88 411.97 188.03
49 10.40 411.97 401.57 198.43
50 10.92 401.57 390.65 209.35
51 11.47 390.65 379.18 220.82
52 12.04 379.18 367.14 232.86
53 12.64 367.14 354.50 245.50
54 13.27 354.50 341.23 258.77
55 13.94 341.23 327.29 272.71
56 14.64 327.29 312.65 287.35
57 15.37 312.65 297.28 302.72
58 16.14 297.28 281.15 318.85 x
59 16.94 281.15 264.21 335.79
60 17.79 264.21 246.42 353.58
61 18.68 246.42 227.74 372.26
62 19.61 227.74 208.12 391.88
63 20.59 208.12 187.53 412.47
64 21.62 187.53 165.91 434.09
65 22.70 165.91 143.20 456.80
66 23.84 143.20 119.36 480.64
67 25.03 119.36 94.33 505.67
68 26.28 94.33 68.05 531.95
69 27.60 68.05 40.45 559.55
70 28.98 40.45 11.47 588.53
Initial= 600.00 Annual Growth Rate= 0.03 + discovery
Year Rate start end Total used
1 1.00 600.00 599.00 1.00
2 1.03 599.00 597.97 2.03
3 1.06 597.97 596.91 3.09
4 1.09 596.91 595.82 4.18
5 1.13 595.82 594.69 5.31
6 1.16 594.69 593.53 6.47
7 1.19 593.53 592.34 7.66
8 1.23 592.34 591.11 8.89
9 1.27 591.11 589.84 10.16
10 1.30 589.84 588.54 11.46
11 1.34 588.54 587.19 12.81
12 1.38 587.19 585.81 14.19
13 1.43 585.81 584.38 15.62
14 1.47 584.38 582.91 17.09
15 1.51 582.91 581.40 18.60
16 1.56 581.40 579.84 20.16
17 1.60 579.84 578.24 21.76
18 1.65 578.24 576.59 23.41
19 1.70 576.59 574.88 25.12
20 1.75 574.88 573.13 26.87
21 1.81 573.13 571.32 28.68
22 1.86 571.32 569.46 30.54
23 1.92 569.46 567.55 32.45
24 1.97 567.55 565.57 34.43 x
25 2.03 565.57 563.54 36.46
26 2.09 563.54 561.45 38.55
27 2.16 561.45 559.29 40.71
28 2.22 559.29 557.07 42.93
29 2.29 557.07 554.78 45.22
30 2.36 554.78 552.42 47.58
31 2.43 552.42 550.00 50.00
32 2.50 550.00 547.50 52.50
33 2.58 547.50 544.92 55.08
34 2.65 544.92 542.27 57.73
35 2.73 542.27 539.54 60.46
36 2.81 539.54 536.72 63.28
37 2.90 536.72 533.83 66.17
38 2.99 533.83 530.84 69.16
39 3.07 530.84 527.77 72.23
40 3.17 527.77 524.60 75.40
41 3.26 524.60 521.34 78.66
42 3.36 521.34 517.98 82.02
43 3.46 517.98 514.52 85.48
44 3.56 514.52 510.95 89.05
45 3.67 510.95 507.28 92.72
46 3.78 507.28 503.50 96.50
47 3.90 503.50 499.60 100.40
48 4.01 1099.60 1095.59 104.41 x Discover 600 Units
49 4.13 1095.59 1091.46 108.54
50 4.26 1091.46 1087.20 112.80
51 4.38 1087.20 1082.82 117.18
52 4.52 1082.82 1078.30 121.70
53 4.65 1078.30 1073.65 126.35
54 4.79 1073.65 1068.86 131.14
55 4.93 1068.86 1063.93 136.07
56 5.08 1063.93 1058.85 141.15
57 5.23 1058.85 1053.61 146.39
58 5.39 1053.61 1048.22 151.78
59 5.55 1048.22 1042.67 157.33
60 5.72 1042.67 1036.95 163.05
61 5.89 1036.95 1031.05 168.95
62 6.07 1031.05 1024.99 175.01
63 6.25 1024.99 1018.74 181.26
64 6.44 1018.74 1012.30 187.70
65 6.63 1012.30 1005.67 194.33
66 6.83 1005.67 998.84 201.16
67 7.03 998.84 991.80 208.20
68 7.25 991.80 984.56 215.44
69 7.46 984.56 977.09 222.91
70 7.69 977.09 969.41 230.59
71 7.92 969.41 961.49 238.51 x
72 8.16 961.49 953.33 246.67
73 8.40 953.33 944.93 255.07
74 8.65 944.93 936.28 263.72
75 8.91 936.28 927.37 272.63
76 9.18 927.37 918.19 281.81
77 9.45 918.19 908.74 291.26
78 9.74 908.74 899.00 301.00
79 10.03 899.00 888.97 311.03
80 10.33 888.97 878.64 321.36
81 10.64 878.64 868.00 332.00
82 10.96 868.00 857.04 342.96
83 11.29 857.04 845.75 354.25
84 11.63 845.75 834.12 365.88
85 11.98 834.12 822.14 377.86
86 12.34 822.14 809.81 390.19
87 12.71 809.81 797.10 402.90
88 13.09 797.10 784.01 415.99
89 13.48 784.01 770.54 429.46
90 13.88 770.54 756.65 443.35
91 14.30 756.65 742.35 457.65
92 14.73 742.35 727.62 472.38
93 15.17 727.62 712.45 487.55
94 15.63 712.45 696.82 503.18
95 16.10 696.82 680.73 519.27 x
96 16.58 680.73 664.15 535.85
97 17.08 664.15 647.07 552.93
98 17.59 647.07 629.49 570.51
99 18.12 629.49 611.37 588.63
100 18.66 611.37 592.71 607.29
101 19.22 592.71 573.49 626.51
102 19.80 573.49 553.70 646.30
103 20.39 553.70 533.31 666.69
104 21.00 533.31 512.31 687.69
105 21.63 512.31 490.68 709.32
106 22.28 490.68 468.40 731.60
107 22.95 468.40 445.45 754.55
108 23.64 445.45 421.81 778.19
109 24.35 421.81 397.47 802.53
110 25.08 397.47 372.39 827.61
111 25.83 372.39 346.56 853.44
112 26.60 346.56 319.96 880.04
113 27.40 319.96 292.56 907.44
114 28.22 292.56 264.34 935.66
115 29.07 264.34 235.27 964.73
116 29.94 235.27 205.32 994.68
117 30.84 205.32 174.48 1025.52
118 31.77 174.48 142.72 1057.28 x
119 32.72 142.72 110.00 1090.00
120 33.70 110.00 76.30 1123.70
121 34.71 76.30 41.59 1158.41
122 35.75 41.59 5.84 1194.16