Scaling Laws Math Example 4

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Example 4

medium
A population model gives P(t)=P0ertP(t) = P_0 e^{rt}. If rr is doubled (growth rate doubles), by what factor does P(T)P(T) change for a fixed time TT?

Solution

  1. 1
    Original: P(T)=P0erTP(T) = P_0 e^{rT}.
  2. 2
    Doubled rate: P′(T)=P0e2rT=P0(erT)2P'(T) = P_0 e^{2rT} = P_0 (e^{rT})^2.
  3. 3
    Ratio: P′(T)P(T)=P0e2rTP0erT=erT\frac{P'(T)}{P(T)} = \frac{P_0 e^{2rT}}{P_0 e^{rT}} = e^{rT}.
  4. 4
    So doubling the rate multiplies the population at time TT by erTe^{rT}.

Answer

P′(T)=erT⋅P(T) (multiplied by erT)P'(T) = e^{rT} \cdot P(T) \text{ (multiplied by } e^{rT}\text{)}
Exponential growth does not scale linearly with its parameters. Doubling the rate replaces erTe^{rT} with e2rT=(erT)2e^{2rT} = (e^{rT})^2 — a squaring, not a doubling, of the growth factor.

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.

Learn more about Scaling Laws →

More Scaling Laws Examples

Example 1 easy

If you scale a square's side length by a factor of 3, by what factor does the area change? State the

Example 2 medium

A sphere of radius [formula] has volume [formula]. If the radius doubles, by what factor does the vo

Example 3 easy

If the radius of a circle doubles, by what factor does the circumference change? What about the area

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