An important tool in archeological research is radiocarbon dating, developed by

Question

An important tool in archeological research is radiocarbon dating, developed by the American chemistWillard F. Libby. This is a means of determining the age of certain wood and plant remains, hence of animal or human bones or artifacts found buried at the same levels. Radiocarbon dating is based on the fact that some wood or plant remains contain residual amounts of carbon-14, a radioactive isotope of carbon. This isotope is accumulated during the lifetime of the plant and begins to decay at its death. Since the half-life of carbon-14 is long (approximately 5730 years2), measurable amounts of carbon-14 remain after many thousands of years. If even a tiny fraction of the original amount of carbon-14 is still present, then by appropriate laboratory measurements the proportion of the original amount of carbon-14 that remains can be accurately determined. In other words, if Q(t) is the amount of carbon-14 at time t and Q0 is the original amount, then the ratio Q(t)/Q0 can be determined, at least if this quantity is not too small. Present measurement techniques permit the use of this method for time periods of 50,000 years or more. Suppose that certain remains are discovered in which the current residual amount of carbon-14 is 5% of the original amount. Determine the age of these remains.

Explanation / Answer

a.) The solution of a differential equation of this type has the form: Q(t)=ae^-rt. Q(0) = ae^0= a is the original amount present. Since the half life is 5730, this means that
Q(5730) = 0.5a = ae^(-5730r), or 0.5 = e^-5730t. Taking the natural log of both sides: ln(0.5) = -5730t, or
r = -ln(.5)/5730 = 1.20968094E-4

b.) Q(t) = (Q0)e^-1.20968094E-4t

c.) Q(t)=(0.2)Q0 = (Q0)e^-1.20968094E-4t
ln(0.2) = ^-1.20968094E-4t
t = -ln(0.2)/1.20968094E-4t ? 13305 years (rounded to the nearest year)

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