Question

Element X decays radioactively with a half life of 12 minutes. If there are 200

a
grams of Element X, how long, to the nearest tenth of a minute, would it take
the element to decay to 20 grams?

Answer 1

`\textit{Amount for Exponential Decay using Half-Life} \\\\ A=P\left( (1)/(2) \right)^{(t)/(h)}\qquad \begin{cases} A=\textit{current amount}\dotfill &20\\ P=\textit{initial amount}\dotfill &200\\ t=\textit{elapsed time}\\ h=\textit{half-life}\dotfill &12 \end{cases} \\\\\\ 20=200\left( (1)/(2) \right)^{(t)/(12)}\implies \cfrac{20}{200}=\left( (1)/(2) \right)^{(t)/(12)}\implies \cfrac{1}{10}=\left( (1)/(2) \right)^{(t)/(12)}`

`\log\left( \cfrac{1}{10} \right)=\log\left[ \left( (1)/(2) \right)^{(t)/(12)} \right]\implies \log\left( \cfrac{1}{10} \right)=t\log\left[ \left( \sqrt[12]{(1)/(2)} \right) \right] \\\\\\ \cfrac{\log\left( (1)/(10) \right)}{\log\left[ \left( \sqrt[12]{(1)/(2)} \right) \right]}=t\implies \implies \stackrel{mins}{39.9}\approx t`