Radioactive decay with half-life

Text

A radioactive substance has initial mass $80$ g and halves every $5$ years.

• (a) write the law $y=b\cdot a^{x}$ of mass versus time $x$ (years);
• (b) compute the mass after $12$ years;
• (c) after how many years does the mass fall to $5$ g? (use $\log_{10}2\approx0{,}301$)

Solution

(a) $b=80$ and, from halving in $5$ years, $a=\left(\tfrac12\right)^{1/5}=2^{-1/5}\approx0{,}8706$: $\;y=80\cdot\left(\tfrac12\right)^{x/5}$.

(b) $y(12)=\dfrac{80}{2^{2,4}}\approx 15{,}15$ g.

(c) $5=80\cdot\left(\tfrac12\right)^{x/5}\Rightarrow \left(\tfrac12\right)^{x/5}=\tfrac{1}{16}=\left(\tfrac12\right)^4\Rightarrow x=20$ years.

$\boxed{y=80\cdot\left(\tfrac12\right)^{x/5};\ \ y(12)\approx15{,}15\ \text{g};\ \ x=20\ \text{years}}$