Question

I need to know q2 please help

Answer 1

Given that the half-life of uranium is
`4.5*10^9` years, you know that after that amount of time you're left with half as much uranium as what you start with:


`\frac12=e^(-k*4.5*10^9)`

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


`\frac12=e^(-k*4.5*10^9)`

`\implies \ln\frac12=\ln e^(-k*4.5*10^9)`

`\implies -\ln2=-k*4.5*10^9\ln e`

`\implies k=(\ln2)/(4.5)*10^(-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


`\frac18=e^(-kt)`

`\implies\ln\frac18=\ln e^(-kt)`

`\implies-\ln8=-kt\ln e`

`\implies t=\frac{\ln8}k=(\ln8)/((\ln2)/(4.5)*10^(-9))=(\log_28)/(4.5)*10^9\approx0.66*10^9`