Copyright © 2011 Pearson Education, Inc. Exponential Astonishment 8 A Discussion Paragraph 1 web 31. Computing Power 32. Web Growth 1 world 33. Linear

Copyright © 2011 Pearson Education, Inc.

Exponential Astonishment

8 A Discussion Paragraph

1 web31. Computing Power32. Web Growth

1 world33. Linear Growth34. Exponential Growth

Copyright © 2011 Pearson Education, Inc. Slide 8-3

Unit 8B

Doubling Time and Half-Life

8-B


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.

8-B


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

double2t T

Doubling Time

8-BDoubling with Compound InterestCN (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?


8-B


Example

World Population Growth: World population doubled from 3 billion in 1960 to 6 billion in 2000. 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 21.25 ≈ 14.3 billion

8-BWorld Population GrowthCN (2a-b)

2. World population doubled from 3 billion in 1960 to 6 billion in 2000. 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?


8-B


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%.

70doubleT

P

8-BPopulation Doubling TimeCN (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.


8-BSolving the Doubling Time FormulaCN (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.


8-B


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

( ) half1 2t T

Exponential Decay and Half-Life

8-BCarbon-14 DecayCN (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?


8-BPlutonium After 100,000 YearsCN (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?


8-B


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%.

70halfT

P

The Approximate Half-Life Formula

8-BDevaluation of CurrencyCN (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?


8-B


Exact Doubling Time and Half-Life Formulas

doublelog 2 log (1 + )r

T

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

halflog 2 log (1 + )r

T

For a exponentially decaying quantity, use a negative value for r. The halving time is

8-BLarge Growth RateCN (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.


8-BRuble RevisitedCN (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.


8-BQuick QuizCN (11)

11. Please answer the quick quiz multiple choice questions 1-10 on p. 497.


8-BHomework8B

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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Copyright © 2011 Pearson Education, Inc. Exponential Astonishment 8 A Discussion Paragraph 1 web 31. Computing Power 32. Web Growth 1 world 33. Linear