Question

the half-life of a radioactive element is 133 days, but your sample will not be useful to you after 90% of the radioactive nuclei originally present have disintegrated. about how many days can you use the sample?

Answer 1

we can do this

Answer 2

since the half life is 133 days you can use
\[A(t)=A_0(\frac{1}{2})^{\frac{t}{133}}\]as a model for how much you have left after t days with initial amount
\[A_0\] now you need 90% = 0.9 so set
\[.9=(\frac{1}{2})^{\frac{t}{133}}\] solve for t via
\[\frac{t}{133}=\frac{\ln(.9)}{\ln(.5)}\]
\[t=\frac{133 \ln(.9)}{\ln(.5)}\] with a calculator

Answer 3

whoa all wrong. after 90% is gone! leaves 10% damn sorry

Answer 4

\[.1=(\frac{1}{2})^{\frac{t}{133}}\]
\[\frac{t}{133}=\frac{\ln(.1)}{\ln(.5)}\]
\[t=\frac{133 \ln(.1)}{\ln(.5)}=441.8\] rounded