Question 946030
Material has half life of t seconds.  How long will it take for s percent of Material sample to decay?


Question is to find how much time until quantity present changes from I to {{{((100-s)/100)*I}}}.


First find k, and then use k to find time to let s percent of sample to decay.


{{{y=I*e^(-kt)}}}
{{{Ie^(-kt)=y}}}
{{{1*e^(-kt)=(1/2)}}}
{{{ln(e^(-kt))=-ln(2)}}}
{{{-kt=-ln(2)}}}
{{{k=ln(2)/t}}}, you know t=30 seconds and want the value...
{{{k=ln(2)/30}}}
{{{highlight(k=0.0231)}}}.


Model Revised:  {{{highlight_green(y=I*e^(-0.0231t))}}}.
Starting with I=1, t is what when {{{y=1-s/100}}} ?
(Your value given for s is 91).
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{{{ln(e^(-0.0231t))=ln(1-s/100)}}}
{{{-0.0231t=ln(1-s/100)}}}
t=-ln(1-s/100)/(-0.0231)
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Easier to avoid sign mistakes if directly using the actual values.
{{{ln(e^(-0.0231t))=ln((100-91)/100)}}}
{{{-0.0231t=ln(9/100)}}}
{{{0.0231t=ln(100/9)}}}
{{{t=ln(100/9)/(0.0231)}}}
{{{highlight(t=104)}}}, seconds.