Question

the half-life of a radioactive element is 138 days, but your sample will not be useful to you after 60% of the radioactive nuclei originally present have disintegrated. about how many days can you use the sample? a. 187 b. 167 c. 172 d. 182

Answer 1

182

Answer 2

\[138 \times 1/(2^{x}) =138 \times 40% \]
\[2^{x} = 100/40\]
\[x = \log_{2} (5/2)\]\[x = 1.321928\]

if x = 1, then x = 138 days,
in this case x = 1.321928, so duration :
138 x 1.321928 = 182.43324

Answer 3

\[(\frac{1}{2})^{\frac{t}{138}}=.6\]
\[\frac{t}{138}=\frac{\ln(.6)}{\ln(.5)}\]
\[t=\frac{138\ln(.6)}{\ln(.5)}\]

Answer 4

oops no that is wrong. after 60% is gone only 40% remains, so should have written
\[t=\frac{138\ln(.4)}{\ln(.5)}\]

Answer 5

Cool how to use that ln?