Exponential Growth / Decay
Growth (k>0) or decay (k<0) over time.
Formula
Variables
Example
A colony of 100 bacteria growing at k = 0.5/h reaches 100·e^(0.5·3) ≈ 448 after 3 hours.
How it works
This formula models situations where the amount changes at a constant percentage rate per unit time, rather than by a fixed amount. The exponential term e^{kt} comes from the fact that the rate of change is proportional to the current amount, so larger amounts grow faster and smaller amounts decay faster. If k is positive, the quantity increases over time; if k is negative, it decreases toward zero.
When to use it
Use this for continuously compounded interest, population growth, radioactive decay, cooling/heating models, and any process where the change depends on how much is already present. It is especially useful when the rate is given as a continuous rate per unit time.
Worked example, step by step
A medicine concentration starts at A_0 = 80 mg and decays with k = -0.12 per hour. Use A = A_0 e^{kt}. A = 80e^{(-0.12)(5)} A = 80e^{-0.6} A \approx 80(0.5488) A \approx 43.9 Final answer: after 5 hours, about 43.9 mg remains.
Common mistakes to avoid
- Using a linear formula instead of exponential change
- Forgetting that k is negative for decay and positive for growth
- Substituting t in the wrong time unit, such as minutes when k is per hour
Did You Know?
Exponential growth is deceptively slow at first then explosive — the reason compound interest and pandemics surprise people.
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