Some amount of a radioactive substance (half-life =10 days ) is spread inside a room and consequently the level of radiation become `50` times the permissible level for normal occupancy of the room. After how many days will the room be safe for occupation?.

Explanation

We have,
`(N)/(N_(0))=((1)/(2))^(t//T)`
Given, half-life `T=10 days`.
`(N)/(N_(0)) =(1)/(50) rArr (1)/(50) =((1)/(2))^(t//T)`
Taking log,
`log 1 -log 50 =(t)/(T) (log 1- log 2)`
`rArr 0-log (100)/(2)=(t)/(T)(0-log -2)`
`rArr log 100 -log2 =(t)/(T)(0-log 2)`
or `log 100 -log 2 =(t)/(T) log 2`
As `log_(10) 2 = 0.3010`, we have
`2 -0.3010 =(t)/(T) xx 0.3010`
or `t=(1.6990)/(0.3010) T=(1699)/(301) xx(10 days)=56.45 days` .