Population Growth

P = P_{0} (1 + r)^{n} or P = P_{0} e^{k t}

Geometric (compound) growth uses a discrete annual rate r: P = P0 (1+r)^n. Exponential growth uses a continuous rate k: P = P0 e^{kt}. They match when k = ln(1+r). Design of water and wastewater facilities often projects 20-year population with a stated r. Doubling time is ln(2)/k or ln(2)/ln(1+r).

$P$: Future population
$P_{0}$: Present population
$r$: Discrete annual growth rate
$n$: Number of years
$k$ (1/yr): Continuous growth rate

Step-by-step solved example

P0 = 100,000, r = 2.0%/yr, n = 10 yr. Find P.

Exponential growth of a population — same shape as a first-order reactor.

1. Compound factor
(1.02)^{10} = 1.21899.
2. Population
P = 100,000 × 1.219 = 121,900.
$P = 121, 900$

Answer: P ≈ 121,900

10 practice questions

0/10 correct

1.P0 = 50,000, r = 0, n = 20. P =

2.Doubling time at r = 2% geometric is nearest

3.k related to discrete r by

4.P0 = 80,000, r = 2.5%/yr, n = 8. P is nearest

5.Arithmetic (linear) growth would instead be

6.A declining city with r = −1%/yr for 10 yr: P/P0 =

7.Per-capita wastewater flow times P estimates

8.e^{0.02×10} vs (1.02)^{10} : the exponential is

9.n for P = 2 P0 at continuous k = 0.035 yr⁻¹ is

10.Logistic growth levels off at a