Growth, decay, and proportional change

Understand the idea

For a constant proportional rate, the solution is exponential. Determine the constant from data using logarithms and interpret its sign and units. Half-life and doubling time describe multiplicative changes, not fixed additive changes. A model in which a temperature difference decays uses that difference as the dependent quantity, producing a shifted exponential.

y(t)=y_0e^{kt},\qquad T_{\text{double}}=\frac{\ln2}{k}\ (k>0),\quad T_{\text{half}}=\frac{\ln2}{|k|}\ (k<0)

Worked example

A population satisfies $\displaystyle P'=kP$ , $\displaystyle P(0)=100$ , and $\displaystyle P(4)=200$ . Find $\displaystyle P(t)$ .

The model is

\displaystyle 100e^{kt}

. Since

\displaystyle 2=e^{4k}

,

\displaystyle k=\ln2/4

. Therefore

\displaystyle P(t)=100e^{(\ln2)t/4}

. The population doubles every four time units under this model.

Try it yourself

A substance has half-life $\displaystyle 6$ hours and starts at $\displaystyle 80$ grams. Find the amount at $\displaystyle 18$ hours.

Show solution and reasoning

Three half-lives pass, so the amount is $\displaystyle 80(1/2)^3=10$ grams. The equivalent model is $\displaystyle 80e^{-(\ln2)t/6}$ .