SME
Human Geography · Problem type
Population density away from the centre
Use this for “A city’s centre has 20,000 people per km² and a density gradient of 0.2 per km. How dense is it 10 km out?”
Chart
D(10) ≈ 2,707
This is an example. Type your own values to start a new problem.
D (people per km²)
=
D₀ (people per km²)
×
e
−
×
Formulas
D
=
D₀
×
e
−
b
×
x
2,706.7057
=
20,000
×
e
−0.2 × 10
Assumptions
•
Clark’s model: density falls by the same fraction for each km from one centre, in every direction.
•
D₀ is the density the curve gives at the centre; offices can leave the real core emptier.
•
A smaller gradient b is a flatter curve: a city spread out by cars and commuter rail.
Step-by-step
WE KNOW
D₀ = 20,000 people per km², b = 0.2 km⁻¹, x = 10 km
FIND
density (D)
Step 1 · Find density (D)
Use D = D₀ × e−b × x
Each
km
out
keeps
the
same
fraction
e
−
b
of
the
density,
so
x
km
keep
e
−
bx
of
it.
D
=
20,000
×
e
−0.2 × 10
0.2 × 10 = 2
e
-2
=
0.1353
D
=
2,706.706
people
per
km²
Check
Put the numbers back in. Both sides should match.
2,706.706
=
20,000
×
e
−0.2 × 10
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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