## Precalculus (6th Edition) Blitzer

The number of days is $325$.
According to the given condition, the exponential model is $A={{A}_{0}}{{e}^{kt}}$ and the half-life of polonium 210 is 140 days. Consider the function: $A\left( t \right)=0.2{{A}_{0}}$ Half-time ${{t}_{\frac{1}{2}}}=140\text{ days}$ Then, the value of k is as shown below: \begin{align} & k=\frac{\ln 2}{{{t}_{\frac{1}{2}}}} \\ & =\frac{\ln 2}{140} \\ & =0.00495 \end{align} And, \begin{align} & A={{A}_{0}}{{e}^{kt}} \\ & 0.2{{A}_{0}}={{A}_{0}}{{e}^{-0.00495t}} \\ & \ln 0.2=-0.00495t \\ & t=-\frac{\ln 2}{0.00495} \end{align} Thus, \begin{align} & t=-\frac{\ln 2}{0.00495} \\ & =325\text{ days} \end{align} Therefore, the number of days is 325.