Radioactive substance has a half life of 18 years. Decay rate remains constant how long will it before 95 percent of the sample has decayed?

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2. Hello, av8or91!

A radioactive substance has a half life of 18 years. Decay rate remains constant.
How long will it be before 95 percent of the sample has decayed?
The half-life function is: .$\displaystyle A \;=\;A_o\,e^{-kt}$
. . where $\displaystyle A_o$ is the intial amount and .$\displaystyle k \:=\:\frac{\ln(2)}{\text{half-life}}$

Our function is: .$\displaystyle A \;=\;A_o\,e^{-\frac{\ln 2}{18}t}$

If 95% of the sample has decayed, then: .$\displaystyle A \,=\,0.05A_o$

We have: .$\displaystyle A_oe^{-\frac{\ln2}{18}t} \:=\:0.05A_o \quad\Rightarrow\quad e^{-\frac{\ln2}{18}t} \:=\:0.05$

Take logs: .$\displaystyle \ln\left(e^{-\frac{\ln2}{18}t}\right) \:=\:\ln(0.05) \quad\Rightarrow\quad -\tfrac{\ln2}{18}t\ln(e) \:=\:\ln(0.05)$

. . $\displaystyle t \:=\:\frac{18\ln(0.06)}{-\ln2} \;=\; 77.79470571 \;\approx\;77.8\text{ years}$