herraviuj

2022-09-07

I am given the half life of a certain object is 1599 years and the amount after 10,000 years is 0.5 grams. How do I find initial quantity?

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Yuliana Griffith

Expert

Let the model be: $A\left(t\right)=P{e}^{rt}$, then you have: $A\left(1599\right)=\frac{P}{2},A\left(10,000\right)=0.5$. Thus: $\frac{P}{2}=P{e}^{1599r}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}r=-\frac{\mathrm{ln}2}{1599}\approx -0.00043$. Thus $A\left(t\right)=P{e}^{-0.00043t}$. Next $0.5=P{e}^{10,000\left(-0.00043\right)}=P{e}^{-4.3}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}P=0.5{e}^{4.3}\approx 36.85$ units.

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