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?

Yuliana Griffith

Beginner2022-09-08Added 6 answers

Let the model be: $A(t)=P{e}^{rt}$, then you have: $A(1599)={\displaystyle \frac{P}{2}},A(10,000)=0.5$. Thus: $\frac{P}{2}}=P{e}^{1599r}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}r=-{\displaystyle \frac{\mathrm{ln}2}{1599}}\approx -0.00043$. Thus $A(t)=P{e}^{-0.00043t}$. Next $0.5=P{e}^{10,000(-0.00043)}=P{e}^{-4.3}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}P=0.5{e}^{4.3}\approx 36.85$ units.

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