Question

The half-life of radium is 1690 years. If 70 grams are present now, how much will be present in 570 years?

Answer 1

Given that

Half life is 1690 years.

Let A(t) = amount remaining in t years

`\begin{gathered} A(t)=A_0e^(kt) \\ \\ \text{ where }A_{0\text{ }}\text{ is the initial amount} \\ \\ k\text{ is a constant to be determined.} \\ \end{gathered}`

SInce A(1690) = (1/2)A0 and A0 = 70

`\begin{gathered} \Rightarrow35=70e^(1690k) \\ \\ \Rightarrow(1)/(2)=e^(1690k) \\ \\ \Rightarrow\ln((1)/(2))=1690k \\ \\ \Rightarrow k=(\ln((1)/(2)))/(1690) \\ \\ \Rightarrow k=-0.0004 \end{gathered}`

So,

`A(t)=70e^(-0.0004t)`
`\Rightarrow A(570)=70e^(-0.0004(570))\approx55.407\text{ g}`

Therefore, the answer is 55.407 g