Cesium 137 (Cs¹³⁷)is a Short-Lived Radioactive Isotope $e^{-\left(\frac{\ln 2}{30}\right) t}$

Multiple Choice

Cesium 137 (Cs¹³⁷) is a short-lived radioactive isotope.It decays at a rate proportional to the amount of itself present and has a half-life of 30 years (i.e., the amount of Cs¹³⁷ remaining t years after A₀ mg of the radioactive isotope is released is given by $e^{-\left(\frac{\ln 2}{30}\right) t}$ ) .As a result of its operations, a nuclear power plant releases Cs¹³⁷ at a rate of 0.12 mg per year.The plant began its operations in 1990, which we will designate as t = 0.Assume there is no other source of this particular isotope.Which of the following differential equations have a solution R(t) , the amount (in mg) of Cs¹³⁷ in t years? (We are assuming R(0) = 0.)

A) $\frac{d R}{d t}=0.12+e^{-\left(\frac{\ln 2}{30}\right) R}$
B) $\frac{d R}{d t}=0.12-e^{-\left(\frac{\ln 2}{30}\right) }+R$
C) $\frac{d R}{d t}=0.12-\frac{\ln 2}{30} R$
D) $\frac{d R}{d t}=0.12 R-\frac{\ln 2}{30}$