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 }(A_(\circ))/(2) \\ \text{Therefore,} \\ (A_(\circ))/(2)=A_(\circ)e^(27.3k) \\ (1)/(2)=e^(27.3k) \\ In(1)/(2)=Ine^(27.3k) \\ In(1)/(2)=27.3k \\ -0.69314718056=27.3k \\ k=(-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* e^(-0.0254t) \\ (300)/(400)=e^(-0.0254t) \\ 0.75=e^(-0.0254t) \\ In0.75=Ine^(-0.0254t) \\ t=(-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*10) \\ A(10)=400e^(-0.254) \\ A(10)=400*0.77569180204 \\ A(10)=310.276720819 \\ A(10)=310 \end{gathered}`