Example: Radioactive decay and carbon dating
Carbon 14, an unstable isotope of carbon, decays extremely slowly to nitrogen and is used in the dating of fossils. The amount of carbon 14 remaining in a sample that originally contained $A$ grams is given by an exponential function of time $t$:
$C(t) = A(0.999879)^t \qquad$
#[where $t$ is time in years.][donde $t$ es tiempo en años.]#
1. After 10,000 years, a sample that originally contained 100g of carbon 14 will still contain
$C(10{,}000) = 100(0.999879)^{10{,}000} \approx 29.8$ g #[carbon][carbono]# 14.
2. If, after 10,000 years, the amount of carbon 14 in a sample is found to be 50 g, how much did it originally contain?
To find the original amount, substitute the given information in the formula:
$50 = A(0.999879)^{10{,}000}$
\\ $A = \dfrac{50}{0.999879^{10{,}000}} \approx 167.7$ g
3. Dating a sample: If a sample originally containing 100g of carbon 14 has decayed to half that amount, how old is the sample?
#[Again, substitute the given information in the formula:][Otra vez, sustituya la información dada en la fórmula:]#
$50 = 100(0.999879)^{t}$
\\ $\dfrac{1}{2} = 0.999879^t$ \t
\\ $\displaystyle t = \log_{0.999879}\left(\frac{1}{2}\right)\approx 5728$ years \gap[40] \t