Answer:
110 minutes.
Explanation:
Let's see the formula of half-life:
[tex]N(t)=N_0\cdot(\frac{1}{2})^{t\text{/h}}.[/tex]Where N(t) is the quantity remaining, N₀ is the initial quantity, t is time, and h is the half-life.
The problem is telling us that there is 10 % remaining material after 366 minutes. This 10 % corresponds to the division of the quantity remaining and the initial quantity, i.e. N(t)/N₀, so the formula is:
[tex]\begin{gathered} \frac{N(t)}{N_0}=(\frac{1}{2})^{t\text{/h}}, \\ 10\%=(\frac{1}{2})^{t\text{/h}}. \end{gathered}[/tex]Remember that 10 % in decimals is the same that 0.10:
[tex]0.10=(\frac{1}{2})^{t\text{/h}}.[/tex]So if we replace the time (in minutes), we will obtain:
[tex]\begin{gathered} 0.10=(\frac{1}{2})^{366\text{/h}}, \\ \log_{\frac{1}{2}}(0.10)=\frac{366}{h}, \\ h=\frac{366}{\log_{\frac{1}{2}}(0.10)}, \\ h=110.17\text{ minutes}\approx110\text{ minutes.} \end{gathered}[/tex]The answer is that the half-life of the radioactive nucleus of the material is 110 minutes.
