Respuesta :
The half-life formula is given by
[tex]N(t)=N_0(\frac{1}{2})^{\frac{t}{tm}}[/tex]where N(t) is the final amount, N_0 is the initial quantity, t is the elapsed time and tmis the half life of the substance.
In our case,
[tex]\begin{gathered} N(3)=10g \\ N_0=1000g \\ t_m=3\text{ minutes} \end{gathered}[/tex]and we need to find t. By substituting these values into the formula, we have
[tex]10=1000(\frac{1}{2})^{\frac{t}{3}}[/tex]which gives
[tex]\begin{gathered} \frac{10}{1000}=(\frac{1}{2})^{\frac{t}{3}} \\ 0.01=(\frac{1}{2})^{\frac{t}{3}} \end{gathered}[/tex]By applying natural logarithm to both sides ,we have
[tex]\ln (0.01)=\frac{t}{3}\ln (\frac{1}{2})[/tex]which gives
[tex]\begin{gathered} -4.605=\frac{t}{3}(-0.693) \\ \text{then} \\ \frac{t}{3}=\frac{-4.605}{-0.693} \\ \frac{t}{3}=6.6438 \end{gathered}[/tex]Therefore, the elapsed time t is given by
[tex]\begin{gathered} t=3\times6.6438 \\ t=19.93\text{ } \end{gathered}[/tex]Therefore, the answer is 19.93 minutes.
