College Algebra, Chapter 5, Review Exercise, Section Review Exercise, Problem 100

Palladium-100 has a half-life of 4 days. After 20 days a sample has been reduced to a mass of 0.375 g.
(a) Find the initial mass of the sample.
(b) Determine a function that models the mass remaining after $t$ days.
(c) Find the mass after 3 days.
(d) How many days will it take so that 0.15g will remain?

a.) Recall the formula for radioactive decay,
$m(t) = m_0 e^{-rt}$ which $\displaystyle r = \frac{\ln 2}{h}$
Where,
$m(t) =$ the mass remaining at time $t$
$m_0 =$ initial mass
$r =$ rate of decay
$t =$ time
$h =$ half life


$\begin{equation} \begin{aligned} 0.375 &= m_0 e^{-r(20)} && \text{where } r = \frac{\ln 2}{h} = \frac{\ln 2}{4}\\ \\ 0.375 &= m_0 e^{-\left( \frac{\ln 2}{4} \right)(20)} && \text{Divide both sides by } e^{-\left( \frac{\ln 2}{4} \right)(20)}\\ \\ m_0 &= \frac{0.375}{e^{-\left( \frac{\ln 2}{4} \right)(20)}}\\ \\ m_0 &= 12g \end{aligned} \end{equation}$

b.) By substituting all the acquired information, the model is represented as
$\displaystyle m(t) = 12 e^{-\left( \frac{\ln 2}{4} \right)(20)}$

c.) If $t = 3$ days, then

$\begin{equation} \begin{aligned} m(3) &= 12 e^{-\left( \frac{\ln 2}{4} \right)(3)}\\ \\ m(3) &= 7.135 g \end{aligned} \end{equation}$

The mass will be $7.135$g after 3 days
d.) if $m(t) = 0.15g$, then

$\begin{equation} \begin{aligned} 0.15 &= 12 e^{-\left( \frac{\ln 2}{4} \right)(t)} && \text{Divide both sides by } 12\\ \\ \frac{0.15}{12} &= e^{-\left( \frac{\ln 2}{4} \right)(t)} && \text{Take ln of both sides}\\ \\ \ln\left( \frac{0.15}{12} \right) &= -\left( \frac{\ln 2}{4} \right)(t) && \text{Recall } \ln e = 1\\ \\ t &= \frac{-\ln \left( \frac{0.15}{12} \right)}{\frac{\ln 2}{4}} && \text{Solve for } t\\ \\ t &= 25.29 \text{ days} \end{aligned} \end{equation}$