SME
Human Geography · Problem type
A city’s growth rate and doubling time
Use this for “A city grew from 1,200,000 to 3,000,000 people in 25 years. What was its yearly growth rate, and how fast does it double?”
Chart
P(25) = 3,000,000
This is an example. Type your own values to start a new problem.
r (%/yr)
=
100
×
ln
P₂
P₁
÷
t (years)
T₂
Doubling time
calculated · years
Formulas
r
=
100
×
ln(
P₂
÷
P₁
)
t
3.6652
=
100
×
ln(3,000,000 ÷ 1,200,000)
25
T₂
=
100
×
ln(2)
r
18.9118
=
100
×
ln(2)
3.6652
Assumptions
•
r is the steady yearly rate, growing continuously, that takes P₁ to P₂ in t years.
•
The doubling time 100 ln 2 ÷ r, about 69.3 ÷ r, holds only while r stays the same.
•
Births, deaths and migrants all count; a negative r is a shrinking city.
Step-by-step
WE KNOW
P₁ = 1,200,000, P₂ = 3,000,000, t = 25 years
FIND
growth rate (r), doubling time (T₂)
Step 1 · Find growth rate (r)
Use r = 100 × ln(P₂ ÷ P₁) ÷ t
The natural log of the growth factor is the total continuous growth; share it over the years, as a percent.
r
=
100
×
ln(3,000,000 ÷ 1,200,000)
25
3,000,000
1,200,000
=
2.5
ln(2.5) = 0.9163
r
=
3.66516
%/yr
Step 2 · Find doubling time (T₂)
Use T₂ = 100 × ln(2) ÷ r
Doubling
is
a
growth
factor
of
2,
so
it
takes
ln
2
of
total
growth
at
r
100
a
year.
T₂
=
100
×
ln(2)
3.66516
T₂
=
100
×
0.6931
3.6652
T₂
=
69.31
3.6652
T₂
=
18.9118
years
Check
Put the numbers back in. Both sides should match.
3.66516
=
100
×
ln(3,000,000 ÷ 1,200,000)
25
✓
18.9118
=
100
×
ln(2)
3.66516
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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