Answer:
[tex]2.82\cdot 10^9 y[/tex]
Explanation:
A radioactive isotope is an isotope that undergoes nuclear decay, breaking apart into a smaller nucleus and emitting radiation during the process.
The half-life of an isotope is the amount of time it takes for a certain quantity of a radioactive isotope to halve.
For a radioactive isotope, the amount of substance left after a certain time t is:
[tex]m(t)=m_0 (\frac{1}{2})^{\frac{t}{\tau}}[/tex] (1)
where
[tex]m_0[/tex] is the mass of the substance at time t = 0
m(t) is the mass of the substance at time t
[tex]\tau[/tex] is the half-life of the isotope
In this problem, the isotope is uranium-235, which has a half-life of
[tex]\tau=7.04\cdot 10^8 y[/tex]
We also know that the amount of uranium left in the rock sample is 6.25% of its original value, this means that
[tex]\frac{m(t)}{m_0}=\frac{6.25}{100}[/tex]
Substituting into (1) and solving for t, we can find how much time has passed:
[tex]t=-\tau log_2 (\frac{m(t)}{m_0})=-(7.04\cdot 10^8) log_2 (\frac{6.25}{100})=2.82\cdot 10^9 y[/tex]
