Question

After a spill radioactive iodine, measurements showed the ambient radiation levels at the site of the spill to be four times the maximum acceptable limit. The level of radiation from an iodine source decreases according to the formula: R(t)= Re^(-0.004t)
How long will it take for the site to reach an acceptable level of radiation? After a spill radioactive iodine, measurements showed the ambient radiation levels at the site of the spill to be four times the maximum acceptable limit. The level of radiation from an iodine source decreases according to the formula: R(t)= Re^(-0.004t)
How long will it take for the site to reach an acceptable level of radiation? @Mathematics

Answer 1

Given the equation:
\[R(t)=R e^{-0.004t}\]
The problem also gives you the fact that:
\[R(0)=4*R(a)\]
where R(0) is the radiation level initially, and R(a) is the acceptable radiation level.

Lastly, we know that:
\[R(0)=R e^{-0.004(0)}=R\]
Combining all of this information:
\[4*R(a)=R \rightarrow R(a)=R/4=R e^{-0.004a}\]
Now just solve the last part of that equation for a to essentially find acceptable t:
\[[R/4=R e^{-0.004a}]*(4/R)\]
\[1=4e^{-0.004a}\]
\[1/4=e^{-0.004a}\]
\[(Note: \ln (e^{n})=n)\]
\[\ln(1/4)=\ln(e^{-0.004a})\rightarrow-1.38629436=-0.004a\]
\[a=-1.38629436/-0.004=346.57359\]
The problem didn't specify a unit of time, but the answer is that it would take about 346.57 units of time for the spill to reach acceptable levels.

Answer 2

Checking the answer:
\[R(t)=R e−0.004t\]
\[R(346.57359)=R e^{−0.004(346.57359)}=R e^{-1.386294}=R*0.25=(1/4)R(0)\]
So this amount of time would create 1/4th the initial levels of radiation, which is what the problem required.