Question

The half-life of Rad-226 is 1820 years. Suppose one sample has 300 mg of Rad-226.(a) How much radium was there 1820 years ago?(b) Define a function that models the amount A of radium (in mg) in terms of the time t(in years).(c) How much radium will there be after 4000 years?

Answer 1

We were given the following information:

`\begin{gathered} Half\text{ }Life=1,820years \\ t_0=300mg \end{gathered}`

We will proceed to solve as shown below:

a)

`\begin{gathered} Half\text{ }Life=1,820years \\ t_0=300mg \\ 1,820\text{ years ago: } \\ t=2\cdot t_0=2\cdot300=600mg \\ t=600mg \\ \\ \therefore1,820\text{ years ago, there was 600mg of Rad-226} \end{gathered}`

b)

`\begin{gathered} Half\text{ }Life=1,820years \\ t_0=300mg \\ t_1=150mg\Rightarrow1,820 \\ t_2=75mg\Rightarrow2(1,820) \\ t_3=37.5mg\Rightarrow3(1,820) \\ A(t)=t_0\cdot((1)/(2))^t \\ \\ \therefore A(t)=t_0\cdot((1)/(2))^t \end{gathered}`

c)

`\begin{gathered} A(t)=t_0\cdot((1)/(2))^t \\ 1period=1,820years \\ t=(4,000)/(1,820)=2.1978 \\ t=2.1978 \\ A(t)=300*((1)/(2))^(2.1978) \\ A(t)=65.3909\approx65.39 \\ A(t)=65.439mg \\ \\ \therefore A(t)=65.39mg \end{gathered}`