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Calculate radioactive decay, depreciation, and substance elimination
Exponential decay occurs when a quantity decreases at a rate proportional to its current value. The amount decreases rapidly at first, then more slowly over time, creating a characteristic decay curve.
Discrete decay (A = A₀(1 - r)ᵗ) occurs at specific time intervals. Continuous decay (A = A₀e^(-kt)) occurs constantly without interruption, like radioactive decay. The continuous model is more accurate for natural processes.
Decay rate (r or k) is the proportion that decays per time unit. Half-life is the time for half the substance to decay. They're related: for continuous decay, half-life = ln(2) / k ≈ 0.693 / k.
Radioactive decay, drug elimination from the body, capacitor discharge, cooling of objects (Newton's Law), atmospheric pressure with altitude, carbon-14 dating, and asset depreciation all follow exponential decay.
In the discrete model A = A₀(1 - r)ᵗ, the decay rate r must be between 0 and 1 (0% to 100%). In the continuous model A = A₀e^(-kt), the constant k can be any positive value.
If you know A₀ at time 0 and A at time t, calculate: r = 1 - (A/A₀)^(1/t) for discrete decay, or k = -ln(A/A₀) / t for continuous decay.
Mathematically, exponential decay approaches zero but never quite reaches it. In reality, once you reach individual atoms or molecules, quantum effects take over and the last particles do eventually decay completely.
Use whatever units match your decay rate. If the rate is per year, use years for time. If it's per hour (like drug half-life in hours), use hours. Keep rate and time units consistent.