Question 997699
{{{y=Ie^(-kt)}}}
Find k using the given half-life.
{{{ln(y)=ln(I)-kt}}}
{{{ln(y)-ln(I)=-kt}}}
{{{ln(I)-ln(y)=kt}}}
{{{kt=ln(I/y)}}}-----will be re-used later
{{{k=(1/t)ln(I/y)}}}
substitute half life information data.
{{{k=(1/3.8)ln(1/(1/2))}}}
{{{k=ln(2)/3.8}}}
{{{k=0.1824}}}
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Refined Model, {{{highlight(y=Ie^(-0.1824t))}}}


The maximum safety level would be some 100%, not sure based on what units.  Ten percent above the safety level would be some  110% of these units for radon gas.  You want to calculate the time to decay from 110%  to 100%.
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{{{system(I=110, y=100)}}}


Look back at the derivation made earlier, and starting from the step,
{{{kt=ln(I/y)}}}
{{{highlight_green(t=ln(I/y)/k)}}}
{{{t=ln(110/100)/(0.1824)}}}
{{{t=ln(11/10)/(0.1824)}}}
{{{highlight(t=0.5225*days)}}}-----------near enough to say 13 hours.