Carbon Dating

Calculus 2 · Axiom Academy

Once an organism dies its carbon-14 decays on a clock you can read — exponential decay turns "how much is left" into "how old is it." An archaeologist pulls a wooden artifact from the dig. Living things hold a steady level of carbon-14 ; at death that supply stops and the C-14 decays with a half-life of 5,730 years , so what's left is a timer — here's how to read it. Slide time forward and watch the carbon-14 drain away. Every 5,730 years exactly half of what's left disappears — that steady, never-quite-zero fade is exponential decay, N = N₀e −kt . Now read the age off the curve Flip the question: the lab measures how much C-14 is left in the sample, and you solve for time. Take logs and the decay law rearranges to an age — t = ln(fraction) ÷ (−k). Why carbon-14 eventually runs out Each half-life cuts what's left in half again — 100%, then 50, 25, 12.5… Drag the half-lives and watch the bars shrink: after about ten, there's so little C-14 left that the method goes dark. One law, three moves: watch it fade , solve for the age , count the half-lives . The same N = N₀e −kt dates a bone, tracks a medical tracer clearing your bloodstream, charts a drug's concentration between doses, and cools your coffee toward room temperature — anywhere the rate of loss is proportional to what's left.

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