Answer:
C. 30,000
Step-by-step explanation:
We know that, the exponential function for decay is,
[tex]y=ae^{rt}[/tex]
where,
y = the amount after time t,
a = initial amount,
r = rate of decay.
The half-life of a unstable isotope is 10,000 years, so
[tex]\Rightarrow \dfrac{1}{2}=1\cdot e^{r\cdot 10000}[/tex]
[tex]\Rightarrow \dfrac{1}{2}=e^{r\cdot 10000}[/tex]
[tex]\Rightarrow \ln \dfrac{1}{2}=\ln e^{r\cdot 10000}[/tex]
[tex]\Rightarrow -\ln 2={r\cdot 10000}\cdot \ln e[/tex]
[tex]\Rightarrow -\ln 2={r\cdot 10000}\cdot 1[/tex]
[tex]\Rightarrow r=\dfrac{-\ln 2}{10000}[/tex]
Now the function becomes,
[tex]y=ae^{\frac{-\ln 2}{10000}\cdot t}[/tex]
Now, only 1/8 of the radioactive parent remains, so
[tex]\Rightarrow \dfrac{1}{8}=1\cdot e^{\frac{-\ln 2}{10000}\cdot t}[/tex]
[tex]\Rightarrow \dfrac{1}{8}=e^{\frac{-\ln 2}{10000}\cdot t}[/tex]
[tex]\Rightarrow \ln \dfrac{1}{8}=\ln e^{\frac{-\ln 2}{10000}\cdot t}[/tex]
[tex]\Rightarrow -\ln 8={\frac{-\ln 2}{10000}\cdot t}\cdot \ln e[/tex]
[tex]\Rightarrow -\ln 8={\frac{-\ln 2}{10000}\cdot t}[/tex]
[tex]\Rightarrow t=\dfrac{-\ln 8\cdot 10000}{-\ln 2}[/tex]
[tex]\Rightarrow t=30,000[/tex]
