+Sustainability

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 = $r_0+\sum _{n=1\to t-1}{r}_0(1+k{)}^n={\displaystyle \frac{r_0}{k}}(1+k{)}^t-1$

$${\displaystyle \frac{r_0}{k}}(1+k{)}^t-1=R$$

$$t=ln[kR/r_0+1]/ln(1+k)ln[kR/r_0+1]/ln(k)$$

References

• Albert Bartlett - YT
• Tom Murphy’s book, blog

Orgs

• Long Now
  • Manual for civilization
• Post carbon institute