22/3338–12FE Reference Handbook 10.4 · Mathematics · First-Order Models

Handbook formula

Exponential Growth and Decay

Same ODE N'=kN used for population, BOD remaining, radioactive tracers, and continuously compounded money. Half-life when k<0 is ln2/|k|.

Initial value
(1/time)
Growth (or −decay) rate
Half-life

Step-by-step solved example

BOD decays with k=0.23 /d. What fraction remains after 3 days?

DOsatDct
Exponential remaining fraction vs time — same decay shape as a sag/BOD curve.
  1. 1. Evaluate

    N/N0=e^{−0.23×3}=e^{−0.69}≈0.50.

  2. 2. Read

    About one half-life (ln2/0.23≈3.0 d).

Answer: ≈ 50% remains

10 practice questions

0/10 correct

1.If k>0 the quantity

2.Half-life for k=0.0693 /yr is

3.Doubling time is

4.e^{−0.693} is nearest

5.Continuous compounding F=Pe^{rt} matches this model with k=

6.After 2 half-lives the remainder is

7.k has units of

8.N=100 e^{−0.1 t} at t=0 is

9.Linearize by plotting

10.If remainder is 37% , kt is nearest