Radioactive decay

Text

A radioactive sample has a half-life of $8$ days and initial mass $200$ mg.

• (a) How much substance remains after $30$ days?
• (b) After how many days does the mass drop to $25$ mg?

Solution

The model is $m(t)=200\cdot\left(\tfrac12\right)^{t/8}$, with $t$ in days. (a) $m(30)=200\cdot\left(\tfrac12\right)^{30/8}=200\cdot 2^{-3.75}\approx 200\cdot 0.0743\approx 14.87\ \text{mg}.$ (b) Impose $200\cdot\left(\tfrac12\right)^{t/8}=25$, i.e. $\left(\tfrac12\right)^{t/8}=\tfrac{25}{200}=\tfrac18=\left(\tfrac12\right)^{3}$, hence $\tfrac{t}{8}=3\Rightarrow t=24$ days. $\boxed{\,m(30)\approx 14.87\ \text{mg},\quad t=24\ \text{days}\,}$