Question

I need to know q2 please help

Answer 1

Given that the half-life of uranium is $4.5\times10^9$ years, you know that after that amount of time you're left with half as much uranium as what you start with:

$\dfrac12=e^{-k\times4.5\times10^9}$

Solve this equation for $k$ to find the decay rate.

$\dfrac12=e^{-k\times4.5\times10^9}$
$\implies \ln\dfrac12=\ln e^{-k\times4.5\times10^9}$
$\implies -\ln2=-k\times4.5\times10^9\ln e$
$\implies k=\dfrac{\ln2}{4.5}\times10^{-9}$

Now you're looking to find the time it takes for the substance to decay to one eighth of its original amount, which is the time $t$ such that

$\dfrac18=e^{-kt}$
$\implies\ln\dfrac18=\ln e^{-kt}$
$\implies-\ln8=-kt\ln e$
$\implies t=\dfrac{\ln8}k=\dfrac{\ln8}{\frac{\ln2}{4.5}\times10^{-9}}=\dfrac{\log_28}{4.5}\times10^9\approx0.66\times10^9$