College Algebra, Chapter 5, 5.5, Section 5.5, Problem 18

How long will it take for $95 \%$ of a sample to decay, if the radium-221 has a half-life of 30 s?

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 Radium-221 has a half-life of 30 s, then



$
\begin{equation}
\begin{aligned}

r =& \frac{\ln 2}{h} = \frac{\ln 2}{30}
&&
\\
\\
0.95 m_0 =& m_0 e^{- \left( \frac{\ln 2}{30} \right) t }
&& \text{Divide each side by } m_0
\\
\\
0.95 =& e^{- \left( \frac{\ln 2}{30} \right) t }
&& \text{Take $\ln$ of each side}
\\
\\
\ln (0.95) =& - \left(\frac{\ln 2}{30} \right) t
&& \text{Recall that } \ln e = 1
\\
\\
t =& \frac{\ln (0.95)}{\displaystyle - \left( \frac{\ln 2}{30} \right)}
&& \text{Solve for } t
\\
\\
t =& 2.22 s
&&

\end{aligned}
\end{equation}
$