Population Explosion

Calculus 2 · Axiom Academy

Build a continuous exponential growth model from a doubling time, then use it to predict the future and solve for when. A bacterial culture starts with 100 cells and doubles every 20 minutes . Find the continuous growth constant k , predict the population after 3 hours, and determine when the culture first reaches one million cells. Nicely done. You built a continuous growth model from scratch and used it to look both forward in size and forward in time. The model: P(t) = P_0 e^ kt describes any quantity whose rate of change is proportional to its current size. Finding k from a doubling time: a doubling in time T forces e^ kT = 2 , so — here per minute. The explosion: 100 cells become in just 3 hours — nine doublings is a factor of 512 . Solving for "when": to find the time to hit a target, set P(t) equal to it and undo the exponential with ; reaching one million takes minutes (about 4.43 hours). The same two moves — match a known data point to pin down k , then evaluate or invert the model — power continuous compound interest, radioactive decay (with k < 0 ), and cooling. Master them once and the applications come for free.

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