A population of bacteria is growing at a rate proportional to its size. If the population doubles every 3 hours, what is the instantaneous rate of change of the population when it reaches 1000 bacteria?

Core Concept

The population follows exponential growth: P(t) = P₀e^(kt). Since it doubles every 3 hours: 2P₀ = P₀e^(3k) → 2 = e^(3k) → 3k = ln(2) → k = ln(2)/3. The rate of change is P'(t) = kP(t). At P(t) = 1000: P'(t) = (ln(2)/3)(1000) = 1000·ln(2)/3 bacteria per hour.