Scaling Laws Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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A sphere of radius

r

has volume

V = \frac{4}{3}\pi r^3

. If the radius doubles, by what factor does the volume increase? Generalise to a scaling factor

k

1
Original volume: $V = \frac{4}{3}\pi r^3$ .
2
Doubled radius: $V' = \frac{4}{3}\pi (2r)^3 = \frac{4}{3}\pi \cdot 8r^3 = 8V$ .
3
Volume increases by a factor of 8 = $2^3$ .
4
General law: scaling radius by $k$ scales volume by $k^3$ .

Answer

V \text{ scales by } k^3;\text{ doubling radius increases volume by } 8

Volume is a 3-dimensional quantity and scales as

k^3

. This explains why a large sphere holds vastly more than a small one: doubling the radius gives 8 times the volume, not 2 times.

About Scaling Laws

Relationships describing how a quantity changes when the size or scale of a system is multiplied by a factor, often expressed as power laws.

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