Question

The half-life of a radioactive substance is 27.3 years.(a) Find the exponential decay model for this substance. A(t)=A_0e^t(b) How long will it take a sample of 400 grams to decay to 300 grams?(c) How much of the sample of 400 grams will remain after 10 years?

Answer 1

Half life is the length of time it will take a given quantity of a substance to disintegrate to half of it original quantity.

a. Exponential decay model

$\begin{gathered} A(t)=A_{\circ}e^{kt} \\ A_{\circ}=\text{initial amount} \\ \text{Initial amount will reduce to }\frac{A_{\circ}}{2} \\ \text{Therefore,} \\ \frac{A_{\circ}}{2}=A_{\circ}e^{27.3k} \\ \frac{1}{2}=e^{27.3k} \\ In\frac{1}{2}=Ine^{27.3k} \\ In\frac{1}{2}=27.3k \\ -0.69314718056=27.3k \\ k=\frac{-0.69314718056}{27.3} \\ k=-0.02539000661 \\ \\ A(t)=A_{\circ}e^{-0.0254t} \end{gathered}$

b.

$\begin{gathered} A(t)=A_{\circ}e^{-0.0254t} \\ 300=400\times e^{-0.0254t} \\ \frac{300}{400}=e^{-0.0254t} \\ 0.75=e^{-0.0254t} \\ In0.75=Ine^{-0.0254t} \\ t=\frac{-0.28768207245}{-0.0254} \\ t=11.3260658446 \\ t=11.33\text{ years} \end{gathered}$

c.

$\begin{gathered} A(t)=A_{\circ}e^{-0.0254t} \\ A(10)=400e^{-0.0254\times10} \\ A(10)=400e^{-0.254} \\ A(10)=400\times0.77569180204 \\ A(10)=310.276720819 \\ A(10)=310 \end{gathered}$