To find the half-life of a radioactive substance given its annual decay rate, we will use the relationship between the half-life and the decay rate.
The half-life [tex]\( T \)[/tex] of a substance is the time required for half of the substance to decay. This can be calculated using the decay rate, denoted as [tex]\( k \)[/tex]. The formula that relates the half-life [tex]\( T \)[/tex] to the decay rate [tex]\( k \)[/tex] is:
[tex]\[ T = \frac{\ln(2)}{|k|} \][/tex]
where:
- [tex]\( \ln(2) \)[/tex] is the natural logarithm of 2, which is approximately 0.693.
- [tex]\( k \)[/tex] is the decay rate.
Given the decay rate is [tex]\( -0.098 \)[/tex] (which represents a 9.8% annual decrease), we can now substitute this value into the formula.
[tex]\[ T = \frac{\ln(2)}{|-0.098|} \][/tex]
Upon performing the calculation, we get:
[tex]\[ T = \frac{0.693}{0.098} \approx 7.072930413876993 \][/tex]
We now round this result to one decimal place:
[tex]\[ T \approx 7.1 \][/tex]
Hence, the half-life of the radioactive substance is approximately [tex]\( 7.1 \)[/tex] years.
The completed table will look as follows:
\begin{tabular}{|l|l|}
\hline
Half-Life & Decay Rate, [tex]$k$[/tex] \\
\hline
7.1 years & [tex]$9.8 \%$[/tex] per year [tex]$=-0.098$[/tex] \\
\hline
\end{tabular}
Therefore, the half-life is [tex]\( 7.1 \)[/tex] years.
