Exponential Equations and Growth/Decay

Linear: grows by a constant amount (add the same each time) [formula] Exponential: grows by a constant factor (multiply by the same each time) [formula] where: - a = initial value (f(0) = a) - b = growth/decay factor - b > 1: growth (increasing) - 0 < b < 1: decay (decreasing) - x = time or number of periods

Population growth: P = P_0 cdot (1 + r)^t (where r = rate, e.g., 0.03 for 3%) Radioactive decay / half-life: A = A_0 cdot left(frac{1}{2}right)^{t/h} (where h = half-life) Doubling: A = A_0 cdot 2^{t/d} (where d = doubling time) Compound interest: A = Pleft(1 + frac{r}{n}right)^{nt} | Factor | Meaning | |---|---| | 1.05 | 5% growth per period | | 0.90 | 10% decay per period | | 2 | Doubling e…

Example: A population of 500 bacteria doubles every 3 hours. How many bacteria after 12 hours?