# An archaeologist unearths the remains of a wooden box, analyzes for the carbon-14 content, and finds that about 93.75% of the carbon-14 initially present has decayed. The half-life of carbon-14 is 5600 years. How old is the box?

Nov 19, 2017

#### Explanation:

1st order decay gives us:

$\textsf{{N}_{t} = {N}_{0} {e}^{- \lambda \text{t}}}$

Taking natural logs of both sides $\textsf{\Rightarrow}$

$\textsf{\ln {N}_{t} = \ln {N}_{0} - \lambda \text{t}}$

$\textsf{\ln \left({N}_{t} / {N}_{0}\right) = - \lambda \text{t}}$

$\textsf{\lambda = \frac{0.693}{t} _ \left(\frac{1}{2}\right) = \frac{0.693}{5600} = 0.00012375 \textcolor{w h i t e}{x} {a}^{- 1}}$

$\therefore$$\textsf{\ln \left(\frac{6.25}{100}\right) = - 0.00012375 t}$

$\textsf{t = \frac{- 5.07517}{-} 0.00012375 = 41 , 011 \textcolor{w h i t e}{x} \text{yr}}$