Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It is commonly modeled with the function $N(t) = N_0 e^{-kt}$, where $N_0$ is the initial quantity, $k$ is the decay constant, and $t$ is time.
congrats on reading the definition of exponential decay. now let's actually learn it.
In exponential decay, the quantity decreases at a rate proportional to its current value.
The decay constant $k$ determines how quickly the quantity decreases; larger values of $k$ result in faster decay.
The half-life of a substance undergoing exponential decay can be calculated using the formula $t_{1/2} = \frac{\ln(2)}{k}$.
Exponential decay can be integrated to find the total amount decayed over a specific time interval.
Common applications include radioactive decay, population decline, and cooling processes.
Review Questions
What is the general formula for modeling exponential decay?
How do you calculate the half-life of a decaying substance?
Explain how integration is used in finding total quantity decayed over time.
Related terms
Exponential Growth: A process where a quantity increases at a rate proportional to its current value, often modeled as $N(t) = N_0 e^{kt}$ with similar variables as exponential decay but with positive $k$.
Half-Life: The time required for a quantity undergoing exponential decay to reduce to half its initial value.
Decay Constant: $k$ in the exponential decay model; it quantifies the rate at which a substance or quantity decays.