# Forgotten Fundamentals of the Energy Crisis - appendix

by Prof. Al Bartlett

## Acknowledgements

A great deal of correspondence and hundreds of conversations with dozens of people over six years have yielded many ideas, suggestions, and facts which I have incorporated here. I offer my sincere thanks to all who have helped.

## 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 ) = r _{0} e ^{k t} = r_{0} 2 ^{t / T2} *
(1)

where *r _{0}* 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

*T*where:

_{2}
*T _{2} *
(yr) = (ln 2)

*/ k » 70 / P*(2)

and where *P*, the percent growth per year, is 100*k*. 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 = r_{0} *{T to 0} e ^{kt} dt = ( r_{0} / 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 = ( r _{0} / 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 / r*(6)

_{0}+ 1 )
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.

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