Why does the death of a living being affect the decay of carbon-14?

Carbon-14 is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. Carbon-14 decays into nitrogen-14 through beta decay:
By emitting a beta particle (an electron, e-) and an electron antineutrino (νe), one of the neutrons in the carbon-14 nucleus changes to a proton and the carbon-14 nucleus becomes the stable (non-radioactive) isotope nitrogen-14.
The equation governing the decay of a radioactive isotope is
$$ N=N_0 e^{-\frac{t}{\tau}}$$
where No is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. On the other hand, the mean-life τ is the average or expected time a given atom will survive before undergoing radioactive decay.
Since the amount of carbon-14 inside a piece of wood or a fragment of bone decrease as the carbon-14 undergoes radioactive decay, measuring the amount of carbon-14 in a sample provides information that can be used to calculate when the animal or plant died. The mean-life of carbon-14 is 8267 years, so the equation above can be rewritten as:
Nevertheless, radioactive decay is a process that takes place inside the nucleus, so nor a change of temperature neither chemical reactions affect radioactive decay. Carbon-14 atoms inside a living being are decaying after and before the living being dies. So why is this method used efficiently to measure when the living being died? How do we know No, the amount of carbon-14 the living being had at the moment it died, if carbon-14 was also decaying when the plant or the animal was alive?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

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