New trends in physics teaching, v.4; The ... - unesdoc - Unesco

New Trends in Physics Teaching IV
APPENDIX
When a quantity such as rate r(t) of consumption of a resource grows a fixed percentage per
year, the growth is exponential.
where Y, 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 wil increase to twice
its initial size in the doubling time T2 where
T2 (yr) = (ln2) / k = 70/P, (Eq. 2)
and where P, the percentage growth per year, is 100k. The total consumption of a resource
between the present (t = 0) and a future time Tis
=l C r(t) dt.
The consumption in a steady period of growth is
C = rol ekTdt
If the known size of the resource is R tonnes, then we can detennine the exponential expiry
time (EET) by finding the time T, at which the total consumption C is equal to R:
We may solve this for the exponential expiry time T, ,
R = (r0/k)(ekT, - 1). 0%. 5)
EET = T, = (l/k) In (kR/v, + 1). (Eq. 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.
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