2. Exponential Astonishment
8 A Discussion Paragraph
1 web
31. Computing Power
32. Web Growth
1 world
33. Linear Growth
34. Exponential Growth
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3. Doubling Time and Half-Life
Unit 8B
Doubling Time and Half-Life
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4. Doubling and Halving Times
The time required for each doubling in exponential growth is called doubling time.
The time required for each halving in exponential decay is called halving time.
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5. Doubling Time
After a time t, an exponentially growing quantity with a doubling time of Tdouble increases in size by a factor of The new value of the growing quantity is related to its initial value (at t = 0) by
New value = initial value x 2t/Tdouble
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6. Doubling with Compound Interest CN (1)
Compound interest produces exponential growth because an interest-bearing account grows by the same percentage each year. Suppose your bank account has a doubling time of 13 years.
1. By what factor does your balance increase in 50 years?
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7. Example
World Population Growth: World population doubled from 3 billion in 1960 to 6 billion in Suppose that the world population continued to grow (from 2000 on) with a doubling time of 40 years. What would be the population in 2050?
The doubling time is Tdouble 40 years. Let t = 0 represent 2000 and the year 2050 represent t = 50 years later. Use the 2000 population of 6 billion as the initial value.
new value = 6 billion x 250 yr/40 yr
= 6 billion x ≈ 14.3 billion
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8. World Population Growth CN (2a-b)
2. World population doubled from 3 billion in 1960 to 6 billion in Suppose that world population continued to grow (from 2000 on) with a doubling time of 40 years.
a. What would the population be in 2030?
b. In 2200?
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9. Approximate Double Time Formula (The Rule of 70)
For a quantity growing exponentially at a rate of P% per time period, the doubling time is approximately
This approximation works best for small growth rates and breaks down for growth rates over about 15%.
Work through several examples comparing perhaps two different countries and their growth rates or two different radioactive decay percentages. As a challenge, have students try to find the rule of thumb for “tripling time.”
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10. Population Doubling Time CN (3a-b)
3. World population was bout 6.0 billions in 2000 and was growing at a rate of about 1.4% per year.
a. What is the approximate doubling time at this growth rate?
b. If this growth rate were to continue, what would world population be in 2030?
Compare to the result in the last example.
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11. Solving the Doubling Time Formula CN (4)
World population doubled in the 40 years from 1960 – 2000.
4. What was the average percentage growth rate during this period?
Contrast this growth rate with the 2000 growth rate of 1.4% per year.
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12. Exponential Decay and Half-Life
After a time t, an exponentially decaying quantity with a half-life time of Thalf decreases in size by a
factor of The new value of the decaying quantity is related to its initial value (at t = 0) by
New value = initial value x (1/2)t/Thalf
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13. Carbon-14 Decay CN (5)
Radioactive carbon-14 has a half-life of about 5700 years. It collects in organisms only while they are alive. Once they are dead, it only decays.
5. What fraction of the carbon-14 in an animal bone still remains 1000 years after the animal has died?
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14. Plutonium After 100,000 Years CN (6)
Suppose that 100 pound of Pu-239 is deposited at a nuclear waste site.
6. How much of it will still be present in 100,000 years?
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15. The Approximate Half-Life Formula
For a quantity decaying exponentially at a rate of P% per time period, the half-life is approximately
This approximation works best for small decay rates and breaks down for decay rates over about 15%.
Work through several examples comparing perhaps two different radioactive decay percentages. As a challenge, have students try to find the rule of thumb for “tripling time.”
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16. Devaluation of Currency CN (7)
Suppose that inflation causes the value of the Russian ruble to fall at a rate of 12% per year (relative to the dollar).
At this rate, approximately how long does it take for the ruble to lose half its value?
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17. Exact Doubling Time and Half-Life Formulas
For more precise work, use the exact formulas. These use the fractional growth rate, r = P/100.
For an exponentially growing quantity, the doubling time is
For a exponentially decaying quantity, use a negative value for r. The halving time is
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18. Large Growth Rate CN (8)
A population of rats is growing at a rate of 80% per month.
8. Find the exact doubling time for this growth rate and compare it to the doubling time found with the approximate doubling time formula.
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19. Ruble Revisited CN (10)
Suppose the Russian ruble is falling in value against the dollar at 12% per year.
10. Using the exact half-life formula, determine how long it takes the ruble to lose half its value.
Compare your answer to the approximate answer found in #7.
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20. Quick Quiz CN (11)
11. Please answer the quick quiz multiple choice questions 1-10 on p. 497.
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21. Homework 8B Discussion Paragraph 8A Class Notes 1-11 P. 488:1-12 1 web
59. National Growth Rates
60. World Population Growth
1 world
61. Doubling Time
62. Radioactive Half-Life
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