# Radioactivity half life carbon dating

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(See In general, whenever a quantity $$x(t)$$ obeys an exponential decay equation $x(t) = C e^,$ where the continuous decay rate $$k$$ is negative, then the quantity $$x$$ has a half-life $$T$$.After any time period of length $$T$$, the quantity $$x$$ decreases by half. As $$k$$ is negative, the factor $$e^$$ decreases from 1 (at $$t=0$$) towards 0 (as $$t$$ approaches $$\infty$$).Let's investigate what happens to the sample over time. Since $$m$$ has a continuous decay rate of $$-0.000121$$, a general solution to the differential equation is $m(t) = C e^,$ where $$C$$ is a constant.

Samples from the past 70,000 years made of wood, charcoal, peat, bone, antler or one of many other carbonates may be dated using this technique.

Therefore there is a time $$t=T$$ such that $e^ = \dfrac.$ We now solve for $$T$$ and obtain \begin k T &= \log_e \dfrac \ &= - \log_e 2, \end so \[ T = - \dfrac \log_e 2. From time $$t=0$$ to time $$t=T$$, the factor $$e^$$ decreases from $$e^0 = 1$$ to $$e^ = \dfrac$$, that is, decreases by half.

Carbon-14 is also passed onto the animals that eat those plants.

After death the amount of carbon-14 in the organic specimen decreases very regularly as the molecules decay.