document.write( "Question 549637: a sample of radioactive material decayed to 33% of its original mass after 8 days. find the half life of the material. \n" ); document.write( "

Algebra.Com's Answer #358155 by KMST(5328) $\"\"$ $\"About$

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The amount $\"A\"$ remaining, after time $\"t\"$ is a fraction of the initial amount $\"A%5B0%5D\"$
\n" ); document.write( " $\"A%2FA%5B0%5D=%281%2F2%29%5E%28t%2Ft%5B0.5%5D%29\"$ , with $\"t%5B0.5%5D\"$ being the half-life.
\n" ); document.write( "(Note: There are many (equivalent) ways to express that relationship/function, but this is the one that seems to make more direct sense: the fraction is $\"1%2F2\"$ multiplied times itself as many times as half-lives have passed).
\n" ); document.write( "So, taking logarithms of both sides
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\n" ); document.write( "For the problem, with 33% remaining, $\"A%2FA%5B0%5D=33%2F100=0.33\"$ , and $\"t=8\"$ (in days)
\n" ); document.write( "So, we can find $\"t%5B0.50%5D%29\"$ (in days), by solving
\n" ); document.write( " $\"log%280.33%29=%288%2Ft%5B0.5%5D%29%28-log%282%29%29\"$ --> $\"-log%280.33%29%2Flog%282%29=%288%2Ft%5B0.5%5D%29\"$ --> $\"-log%282%29%2Flog%280.33%29=%28t%5B0.5%5D%2F8%29\"$ --> $\"t%5B0.5%5D=-8%2Alog%282%29%2Flog%280.33%29=5\"$ (rounded)
\n" ); document.write( "So the half-life is 5 days. \n" ); document.write( "

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