= Solution
<Radiocarbon dating> measures how long carbon-bearing material has been isolated from a carbon reservoir. Atmospheric <cosmic rays> produce secondary <neutrons>, which create <carbon-14> chiefly through
$$
{}^{14}_{7}\mathrm N+n\longrightarrow{}^{14}_{6}\mathrm C+p.
$$
The <carbon-14> enters atmospheric carbon dioxide and then living organisms through carbon exchange, including photosynthesis and food consumption. While exchange continues, the organism's $^{14}\mathrm C/{}^{12}\mathrm C$ ratio tracks the relevant environmental ratio approximately.
After exchange ceases, the remaining <carbon-14> undergoes <beta-minus decay>,
$$
{}^{14}_{6}\mathrm C\longrightarrow{}^{14}_{7}\mathrm N+e^-+\bar\nu_e,
$$
while stable <carbon-12> remains. With <half-life> $T_{1/2}\simeq5730\,\mathrm{yr}$, the <radioactive decay> law gives
$$
N_{14}(t)=N_{14}(0)e^{-\lambda t},\qquad \lambda=\frac{\ln2}{T_{1/2}}.
$$
Thus measuring the isotope ratio $R$, or the <radioactive activity> per fixed amount of carbon, gives
$$
\boxed{t=\frac{T_{1/2}}{\ln2}\ln\frac{R_0}{R}
=\frac{T_{1/2}}{\ln2}\ln\frac{a_0}{a}.}
$$
Here $a=\lambda N_{14}$ is compared at equal carbon content; $R_0,a_0$ describe the appropriate initial reservoir. <Accelerator mass spectrometry> counts the rare <carbon-14> relative to stable carbon directly; a decay-counting method instead measures its <radioactive activity>.
The initial ratio is not perfectly constant. Atmospheric production changes and reservoir differences therefore require <radiocarbon calibration>, often using independently dated tree rings. Contamination and subsequent carbon exchange must also be controlled. The date measures closure of the carbon system, not automatically the date of every later event involving the object. The physical <half-life> is about $5730$ years; conventional radiocarbon ages historically use $5568$ years and must be calibrated before interpretation as calendar dates.<Radiocarbon calibration by Oxford Radiocarbon Accelerator Unit>[Physical versus conventional half-lives and calibration against tree rings]{ref}
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