Exponential function and doubling time - See functional notes.
Finite resource expiration time
resource left = R
Year 1 resource used = \(r_0\)
Resrouce usage growth rate = k
Year 2 total resource used = \(r_0 + r_0 (1 + k)\)
Year t total resource used = \(r_0 + \sum_{n=1→t-1} r_0 (1 + k)^n = \dfrac{r_0}{k} (1 + k)^t - 1 \)
$$\dfrac{r_0}{k} (1 + k)^t - 1 = R$$
$$t = ln[kR/r_0 + 1]/ln(1+k) ~ ln[kR/r_0 + 1]/ln(k)$$
References
- Albert Bartlett - YT
- Tom Murphy’s book, blog
Orgs
- Long Now
- Manual for civilization
- Post carbon institute