Answer
The number of days is $325$.
Work Step by Step
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.
