Respuesta :
Exponential Decay
The rate of decay of radioactive material is proportional to its actual mass. When solving the resulting equation, we get the mathematical model as follows:
[tex]m(t)=m_oe^{-\lambda\mathrm{}t}[/tex]Where mo is the initial mass, λ is a constant, and t is the time.
The half-life time is the time it takes for the initial mass to be halved, i.e., the remaining mass is mo/2. Substituting into the formula for t=14 days:
[tex]\frac{m_o}{2}=m_oe^{-14\lambda}[/tex]Simplifying by mo and solving for λ:
[tex]\begin{gathered} \frac{1}{2}=_{}e^{-14\lambda} \\ \ln \frac{1}{2}=-14\lambda \\ \lambda=\frac{\ln 2}{14} \\ Calculate\colon \\ \lambda=0.04951 \end{gathered}[/tex]Now our model is complete:
[tex]m(t)=m_oe^{-0.04951t}[/tex]Now we are given the initial mass of a sample mo = 500 grams. It's required to calculate the remaining mass after t = 5 weeks = 5*7 = 35 days.
Substitute values:
[tex]\begin{gathered} m(35)=500e^{-0.04951\cdot35} \\ \text{Operating:} \\ m(35)=500\cdot0.17678 \\ \boxed{m(35)=88.39} \end{gathered}[/tex]Approximately 88 grams of the sample will remail after 5 weeks
