Problem
Question Cesium-137 has a half-life of about 30 years. Given this half-life, we can represent its decay with the exponential decay function \[ A=A_{0} e^{\left(\frac{\ln (0.5)}{30}\right) t} \] If we begin with $200 \mathrm{mg}$ of cesium-137, how long will it take for the Cesium to decay to the point where there is only 1 milligram remaining? Round to the closest year. Provide your answer below:
Solution
Step 1 ：Given that the initial amount of Cesium-137 is \(A_0 = 200\) mg, the remaining amount is \(A = 1\) mg, and the decay rate is \(\frac{\ln (0.5)}{30}\).
Step 2 ：Substitute these values into the decay function \(A=A_{0} e^{\left(\frac{\ln (0.5)}{30}\right) t}\) and solve for \(t\).
Step 3 ：By doing this, we find that \(t = 229\).
Step 4 ：Final Answer: It will take approximately \(\boxed{229}\) years for 200mg of Cesium-137 to decay to 1mg.
