Question

The half-life of $\mathrm{Radium}-226$ is $1590\mathrm{years}$. If a sample contains $300\mathrm{mg}$, how many $mg$ will remain after $3000\text{years}$?

Answer 1

Find the number of mg will remain after $3000\text{years}$:

The formula for the half-life is $p=p_0{\left(\frac{1}{2}\right)}^{\frac{t}{1590}}$.

Because $1\mathrm{period}=1590\mathrm{years}$.

Substitute $p_0=300\mathrm{mg}\mathrm{and}t=3000\mathrm{years}$ in the formula:

$\begin{array}{rcl}p& =& 300{\left(\frac{1}{2}\right)}^{\frac{3000}{1590}}\\ & \Rightarrow & p=300{\left(\frac{1}{2}\right)}^{\frac{3000}{1590}}\\ & \Rightarrow & p=300{\left(\frac{1}{2}\right)}^{\frac{100}{53}}\\ \therefore p& =& 81.22\mathrm{mg}\end{array}$

Therefore, the required amount will be $\mathbf{81}\mathbf{.}\mathbf{22}\mathbf{}\mathbf{mg}$.