College Algebra, Chapter 5, 5.5, Section 5.5, Problem 22

An Egyptian mummy's burial cloth is estimated to contain $59 \%$ of the carbon-14 it contained originally. Determine the period that the mummy was buried. (The half-life of carbon-14 is 5730 years.)



Recall the formula for radioactive decay

$m(t) = m_0 e^{-rt}$ in which $\displaystyle r = \frac{\ln 2}{h}$

where

$m(t)$ = mass remaining at time $t$

$m_0$ = initial mass

$r$ = rate of decay

$t$ = time

$h$ = half-life

If the half-life of carbon-14 is 5730 years, then

$\displaystyle r = \frac{\ln 2}{h} = \frac{\ln 2}{5730}$

So,


$
\begin{equation}
\begin{aligned}

0.59 m_0 =& m_0 e^{- \left( \frac{\ln 2}{5730} \right) (t)}
&& \text{Divide each side by } m_0
\\
\\
0.59 =& e^{- \left( \frac{\ln 2}{5730} \right) t}
&& \text{Take $\ln$ of each side}
\\
\\
\ln (0.59) =& - \left( \frac{\ln 2}{5730} \right) t
&& \text{Divide each side by } \frac{-\ln 2}{5730}
\\
\\
t =& - \frac{\ln (0.59)}{\displaystyle \frac{\ln 2}{5730}}
&& \text{Solve for } t
\\
\\
t =& 4361.75 \text{ years}
&&

\end{aligned}
\end{equation}
$