SOLUTION: Tricky exponential decay word problem: Suppose a small quantity of radon gas, which has a half-life of 3.8 days, is accidentally released into the air in a laboratory. If the re

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Question 997699: Tricky exponential decay word problem:
Suppose a small quantity of radon gas, which has a half-life of 3.8 days, is accidentally released into the air in a laboratory. If the resulting radiation level is 10% above the safe level, how long should the laboratory remain vacated? (Hint: To start with, determine what fraction of the "resulting radiation level" is the maximum safe level.)
The laboratory should remain vacated for ____days
Thank you
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
y=Ie%5E%28-kt%29
Find k using the given half-life.
ln%28y%29=ln%28I%29-kt
ln%28y%29-ln%28I%29=-kt
ln%28I%29-ln%28y%29=kt
kt=ln%28I%2Fy%29-----will be re-used later
k=%281%2Ft%29ln%28I%2Fy%29
substitute half life information data.
k=%281%2F3.8%29ln%281%2F%281%2F2%29%29
k=ln%282%29%2F3.8
k=0.1824
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Refined Model, highlight%28y=Ie%5E%28-0.1824t%29%29

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%.
-
system%28I=110%2C+y=100%29

Look back at the derivation made earlier, and starting from the step,
kt=ln%28I%2Fy%29
highlight_green%28t=ln%28I%2Fy%29%2Fk%29
t=ln%28110%2F100%29%2F%280.1824%29
t=ln%2811%2F10%29%2F%280.1824%29
highlight%28t=0.5225%2Adays%29-----------near enough to say 13 hours.