The formula for half-life in exponential decay is $t_{1/2} = \frac{\ln(2)}{k}$, where $k$ is the decay constant.
Half-life is independent of the initial amount of substance present.
In a continuous exponential decay model, the remaining quantity after time $t$ can be calculated using $N(t) = N_0 e^{-kt}$.
For a function describing population or material decay, integrating from 0 to infinity will yield the total area under the curve, representing the overall quantity over time.
Understanding half-life helps in solving problems involving radioactive decay, pharmacokinetics, and other natural processes modeled by differential equations.
Review Questions
What is the relationship between the decay constant $k$ and half-life $t_{1/2}$?
How do you calculate the remaining quantity of a substance after one half-life has elapsed?
What role does integration play in understanding exponential decay models?
Related terms
Exponential Decay: A process where quantities decrease at a rate proportional to their current value.
$e$ (Euler's Number): An irrational number approximately equal to 2.71828, which serves as the base for natural logarithms and is used in modeling continuous growth or decay.
$\ln(x)$ (Natural Logarithm): $\ln(x)$ represents the logarithm of $x$ with base $e$, commonly used in solving equations involving exponential growth or decay.