Radioactive decay question. W(t) = W(0)^(-0.000122t) where W(t) is the number of Carbon atoms left after "t" years. Assume W(0) = 6x10^10. Assume we cannot detect C if there are less than 10^3 atoms. What is the oldest fossil we can date?
+33614 Set W(t) = 103 and solve for t.
103 = 6*1010*e-0.000122t You didn't have an "e" in your expression, but I suspect it should be there, so I've included it.
Divide both sides by 6*1010
10-7/6 = e-0.000122t
Take logs (natural log) of both sides:
ln(10-7/6) = -0.000122t
Divide both sides by -0.000122
t = -ln(10-7/6)/0.000122 = (7*ln(10)+ln(6))/0.000122
$${\mathtt{t}} = {\frac{\left({\mathtt{7}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{10}}\right)}{\mathtt{\,\small\textbf+\,}}{ln}{\left({\mathtt{6}}\right)}\right)}{{\mathtt{0.000\: \!122}}}} \Rightarrow {\mathtt{t}} = {\mathtt{146\,802.091\: \!149\: \!068\: \!645\: \!811}}$$
t ≈ 146802 years
+33614 Set W(t) = 103 and solve for t.
103 = 6*1010*e-0.000122t You didn't have an "e" in your expression, but I suspect it should be there, so I've included it.
Divide both sides by 6*1010
10-7/6 = e-0.000122t
Take logs (natural log) of both sides:
ln(10-7/6) = -0.000122t
Divide both sides by -0.000122
t = -ln(10-7/6)/0.000122 = (7*ln(10)+ln(6))/0.000122
$${\mathtt{t}} = {\frac{\left({\mathtt{7}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{10}}\right)}{\mathtt{\,\small\textbf+\,}}{ln}{\left({\mathtt{6}}\right)}\right)}{{\mathtt{0.000\: \!122}}}} \Rightarrow {\mathtt{t}} = {\mathtt{146\,802.091\: \!149\: \!068\: \!645\: \!811}}$$
t ≈ 146802 years
