APPENDIX 

When a quantity such as the rate  r( t )  of consumption of a resource grows a fixed percent per year, the growth is exponential:                                                                                        

r ( t )   =   r₀  e ^{k t}  =   r₀  2 ^{t / T2}                                                (1) 

where  r₀  is the current rate of consumption at  t  =  0, e is the base of natural logarithms, k  is the fractional growth per year, and  t  is the time in years.  The growing quantity will increase to twice its initial size in the doubling time T₂where: 
T₂ (yr)  =  (ln 2) / k  »  70 / P                                                    (2) 

and where  P, the percent growth per year, is 100k.  The total consumption of a resource between the present (t  =  0) and a future time T is: 

C  =  {T  to  0}      r(t) dt                                                                 (3) 

The consumption in a steady period of growth is: 

C  =  r₀ {T  to  0}    e ^kt dt =  ( r₀ / k ) ( e ^kt - 1 )                       (4)

If the known size of the resource is R tons, then we can determine the exponential expiration time (EET) by finding the time  T_e  at which the total consumption C is equal to R: 

R  =  ( r₀ / k ) ( e ^kTe - 1 )                                                             (5) 

We may solve this for the exponential expiration time T_e where: 

EET  =  T_e  =  ( 1 / k )  ln ( k R / r₀   + 1 )                                    (6) 

This equation is valid for all positive values of  k  and for those negative values of  k  for which the argument of the logarithm is positive. 

REFERENCES 

1. This paper is based on a series of articles, "The Exponential Function" which is appearing in The Physics Teacher:  

(a) Vol. 14, p. 393 (Oct. 1976); (b) Vol. 14, p. 485 (Nov. 1976);  
(c) Vol. 15, p. 37 (Jan. 1977); (d) Vol. 15, p. 98 (Mar. 1977);  
(e) Vol. 15, p. 225 (Apr. 1977); (f) Vol. 16, p. 23 (Jan. 1978);  
(g) Vol. 16, p. 92 (Feb. 1978); (h) Vol. 16, p. 158 (Mar. 1978).   

An early version of this paper was presented at the Third Annual UMR-MEC Conference on Energy, held at the University of Missouri at Rolla, Oct. 12-14, 1976, and appears in the volume of the Proceedings of the Conference.  The early version, or minor revisions of it have been published in Not Man Apart  published by Friends of the Earth: July / Aug. 1977, Vol. 7, No. 14 pp. 12-13; The Vermillion Flycatcher (Tucson, Arizona Audubon Society, December 1977);  The Colorado Business Review (Grad. Sch. of Business Admin. of the University of Colorado, Jan / Feb 1978). 

2. Newsweek, Dec. 6, 1976. 

3. A. A. Bartlett; Civil Engineering, Dec. 1969,  p. 71. 

4. Time, April 25, 1977, p. 27. 

5. W. Von Engelhardt, J. Goguel, M. King Hubbert, J. E. Prentice, R.A. Price, and R. Trumpy; Environmental Geology, Vol. 1, 193-206 (1976). 

6. A. A. Bartlett, Proceedings of the Third Annual UMR-MEC Conference on Energy,  University of Missouri at Rolla, Missouri, October 12-14, 1976, p. 10. 

7. U.S. Energy Resources, a Review as of 1972, a background paper prepared at the request of the Hon. Henry M. Jackson, Chairman of the Committee on Interior and Insular Affairs of the United States Senate, pursuant to Senate Resolution 45:  M. King Hubbert, A National Fuels and Energy Policy Study, Serial 93-40 (92-75) Part 1 (U.S. GPO, Washington, D.C., 1973), $2.35, 267 pages.  This document is an invaluable source of data on consumption rates and trends in consumption, for both the U.S.A. and the world.  In it Hubbert also sets forth the simple calculus of his methods of analysis.  He does not confine his attention solely to exponential growth.  He predicts that the rate of rise and subsequent fall of consumption of a resource will follow a symmetrical curve that looks like the normal error curve.  Several figures in this paper are redrawn from Hubbert's paper. 

8. L. Ruedisili and M. Firebaugh, Perspectives on Energy, (Oxford University Press, New York,  1975).  
  
9. M. King Hubbert, Resources and Man, National Academy of Sciences and National Research Council, (Freeman, San Francisco, 1969), Chapter 8. 

10.  M. King Hubbert, "Energy Resources of the Earth," Scientific American, Sept. 1971, p. 60. Reprinted as a book (Freeman, San Francisco, 1971). 

11. M. Iona, Physics Teacher, Vol. 15, p. 324 (1977). 

12. Emile Benoit, "The Coming Age of Shortages," Bulletin of Atomic Scientists, January 1976, p. 7.  Benoit attributes his information to David Pimintel et al., "Food Production and the Energy Crisis," Science, Vol. 182, p. 448 (Nov. 2, 1973).  This article is the first of three by Benoit (Bulletin of Atomic Scientists, Jan., Feb., Mar., 1976,).   These are one of the best presentations I have read of coming problems of food, fuels, and resources. 

13. Newsweek, Jan. 31, 1977. 

14. "Factors Affecting the Use of Coal in Present and Future Energy Markets" a background paper prepared by The Congressional Research Service at the request of Sen. Henry M. Jackson, Chairman of the Committee on Interior and Insular Affairs of the United States Senate pursuant to Senate Resolution 45, a National Fuels and Energy Policy Study Serial No. 93-9 (92-44) (U.S. GPO, Washington, D.C., 1973) pp. 41, 42, 15. 

15. "The Energy Crisis" a booklet by the U.S. Energy Research and Development Agency  (ERDA) Oak Ridge, Tennessee, no date, p. 3. (1975 or 1976). 

16. Associated Press story "Energy Head Stresses Coal Reserves," in the Boulder Daily  Camera, July 5, 1975. 

17. "America's Coal: A Gold Mine of Energy," Exxon Corporation two-page full-color ad in Newsweek, 1975. 

18. "They're trying to tell us something.  We're foolish not to listen," American Electric Power Company, Inc.  Two-page ad in Newsweek, 1975. 

19. "The call to greater energy independence" American Electric Power Company, Inc., ad in Newsweek, Nov. 3, 1975. 

20. "An open letter on energy to those who are still employed." American Electric Power Company, Inc., ad in Newsweek, Jan. 12, 1976. 

21. W. H. Miernyk, Journal of Energy and Development, Vol. 1, No. 2, p. 223 (1976). 

22. "The Whale Oil, Chicken, and Energy Syndrome," an address before the Economic Club of Detroit by Walter B. Wriston, Chairman, First National City Corporation, Feb. 25, 1974. 

23. "The Transitional Storm, Part I, An Explanation," by the Edison Electric Institute for the Electric Companies, in  Broadcasting, July 26, 1976. 

24. Charles O. Frush, "Moral Basis for Mineral Resource Use and Development Policy"  The Mines Magazine, Colorado School of Mines, March 1973, p. 20. 

25. J. C. Fisher, "Physics and the Energy Problem," Physics Today, American Institute of Physics, New York, 1974. 

26. "Opening Remarks, UMR-MEC Conference on Energy," R. L. Bisplinghoff, Proceedings of the Conference, Oct. 7-9, 1975, University of Missouri at Rolla. 

27. Washington Star, Feb. 12, 1977. 

28. L. G. Hauser, "Creating the Electric Energy Economy," Proceedings of the Second Annual UMR-MEC Conference on Energy,  October 7-9, 1975, p. 3., University of Missouri at Rolla. 

29. Gil Bailey, "Conservation–Development Proposed As Solution," Washington Bureau of the Boulder Daily Camera, March 13, 1973. 

30. Time, May 19, 1975, p. 55. 

31. D. Brower, Not Man Alone, Vol. 6, No. 20, Nov. 1976; Friends of the Earth, 529 Commercial, San Francisco. 

32. C. C. Garvin, Jr., Chairman of the Exxon Corporation; Full page ad in the Rocky Mountain News, July 23, 1976. 

33. G. Pazik, in a special editorial feature, "Our Petroleum Predicament," in Fishing Facts ("The magazine for today's freshwater fisherman"), Northwoods Publishing Co., P.O. Box 609, Menomonee Falls, WI 53051.  Nov. 1976.  Reprints are available at $0.30 each from the publisher.  This is an excellent summary of the present situation and of the way we got into our petroleum predicament. 

34. The Arizona Republic, Feb. 8, 1978. 

35. "Conservation is like Cholesterol" an ad copyrighted 1976 by the Mobil Oil Corporation. 

36. Boulder Daily Camera, April 4, 1977. 

37. Boulder Daily Camera, May 16, 1977. 

38. U.S. News & World Report, July 25, 1977, P. 8. 

39. Boulder Daily Camera, July 10, 1977. 

40. Amory Lovins, "Energy Strategy, the Road Not Taken," Foreign Affairs, Oct. 1976.  This material is now available as a book, Soft Energy Paths; Toward a Durable Peace Ballinger, Cambridge, MA, 1977).  It is said that this book "could very well be the most important book on energy policy of this decade." 

41. W. L. Rogers, Special Assistant to the Secretary of the Interior, quoted in the Denver Post, Nov. 19, 1976. 

42. Time, April 4, 1977, p. 63. 

43. Technology Review, Dec. 1976, p. 21, reprinted in the second edition of Ref. 8. 

44. Robert H. Romer, Energy–An Introduction to Physics (Freeman, San Francisco, 1976), pp. 594-597.  In addition to making energy the central theme of an introductory text, this book has 18 appendices (61 pages) of data ranging from  "Units and conversion factors" to the "History of energy production and consumption in the world and in the United States" to "Exponential growth" to "Consumer prices of common sources of energy."  The book is at once a text and a valuable source of reference data. 

45. Melvin Laird, "The Energy Crisis: Made in U.S.A." Reader's Digest, Sept. 1977, P. 56. 

46. M. Stanton Evans, Clear and Present Dangers, (Harcourt Brace Jovanovich, New York, 1975). 
 
REPRINTINGS

This paper has been rewritten and reprinted many times in the 20 years since it was first published.

The paper was enlarged and was published in: Mineral & Energy Resources, Colorado School of Mines, Golden, Colorado ;Part I, Vol. 22, Sept. 1979, pp. 1-46; Part II, Vol. 22, Nov. 1979, pp. 1-9; Part III, Vol. 23, Jan. 1980, pp. 1-10.

The enlarged version was also published in the Journal of Geological Education, Vol. 28, Jan. 1980, pp. 4-35.

The paper was rewritten as a chapter in the book, Perspectives on Energy  by L.C. Ruedisili and M.W. Firebaugh, Third Edition, Oxford University Press, New York City, 1982.

The paper was reprinted in New Trends in Physics Teaching, Vol. IV, 1984, pp. 20-37 by the United Nations Educational Scientific and Cultural Organization in Paris, France.

Short versions of this paper have been printed as essays in introductory physics textbooks by Halliday & Resnick, Serway, and Tipler.  Other authors of physics texts have written chapters or sections in their texts using these applications of exponential arithmetic.

The paper has been reprinted in full or abridged in over  30  different publications or proceedings, and was translated into Spanish for publication in Mexico.

I adapted the paper to data on energy in Canada, and it was published as  "Forgotten Fundamentals of the Energy Crisis: A Canadian Perspective," by the Industrial Energy Division of the Ministry of Energy, Mines, and Resources of the Federal Government of Canada, Ottawa, Canada, May 1986.  

This paper was listed as one of ten "memorable papers" for the year 1978 that was included in a list of  "Memorable papers from the American Journal of Physics, 1933-1990"  R.H. Romer, American Journal of Physics, Vol. 59, March 1991, p. 205.

The paper was included in the "Physics Teachers' CD-ROM Toolkit" published by the University of Nebraska, 1993.

Video copies of Dr. Bartlett's lecture are available from University of Colorado Television; Academic Media Services; Campus Box 379; Boulder, CO 80309-0379; (303) 492-1857. 

Reprinted with permission from The American Journal of Physics,  Volume 46, September 1978, pages 876 to 888.  Copyright 1978 American Association of Physics Teachers.