Quantity of Radio Isotope

Given

• $\mbox{A radioactive isotope has a half-life of} \; 18 \; \mbox{days} $
• $\mbox{You wish to have 25 grams after 33 days.} $

Question

• $\mbox{How much radioactive isotope should you start with?}$

Rationale

• $\dfrac{dm}{dt} = - k * m $
• $ \dfrac{dm}{m} = - k * dt    $
• $\int \dfrac{dm}{m} = - k \int dt    $ 
• $\ln{(m)}  = - k t   +  C$
• $ m = e^{-kt+C} = C e^{-kt} = m_0 e^{-kt}$
• $\mbox{Substituting the half-life conditions}  $
• $\dfrac{1}{2} m_0 = m_0 e^{-18k}$
• $\dfrac{1}{2} = e^{-18k}$
• $2 = e^{18k} $
• $\ln(2) = 18k$
• $k = \dfrac{\ln(2)}{18}$
• $\mbox{Updating our decay equation for k} $
• $ m = m_0 e^{-\dfrac{\ln(2)}{18}t}$
• $\mbox{Determine answer to question}  $
• $ 25 = m_0 e^{-\dfrac{\ln(2)}{18}33}$
• $ m_0 = 25  e^{\dfrac{\ln(2)}{18}33} = 89.08987181 \approx 89.1$