Population Growth Model

Calculus 2 · Axiom Academy

One relationship — a population grows in proportion to its own size — runs every bacterial culture, every tumor, every unchecked species. Put it in your hands. A culture starts with 100 cells . The more cells there are, the more divide each minute, so the growth rate is proportional to the population: dP/dt = kP , which solves to P = A·e kt . Three things you can do with one model. Set a growth rate k , hit Go, and watch the count climb in real time. Each cell that appears makes the next batch appear faster — that runaway curve is P = A·e kt , live. When will it reach your target? Flip the question around. You need a certain number of cells — so set the target and read off the time it takes. Same model, rearranged with a logarithm: t = ln(target ÷ A) ÷ k . Here's the number microbiologists actually quote: how long to double . Drag the rate and watch the doubling time ln 2 ÷ k shrink — the ×2 marks pack tighter, and the same window buys far more doublings. One model, three moves: feel it , solve it , weigh it . Anywhere a quantity grows in proportion to its own size — bacteria in a flask , tumor cells , an invasive species , even compound interest — the same P = A·e kt describes it, and ln 2 ÷ k is its doubling time, no matter how many you started with.

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