An important tool in archaeological research is the radiocarbon dating developed by the American chemist Willard F. Libby. This is the means of determining the age of certain wood and plant remains, and 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 years), measure 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 maount 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 Qo is the original amount, then the ration Q(t)/Qo can be determined, as long as this quantity is not too small. Present measurement techniques permite the use of this method for time periods of 50,000 years or more. (a) Assuming the Q satisfies the differential equation Q' = -rQ, determine the decay constant r for carbon-14 (hint: interpret the paragraph above). (b) Find an expression for Q(t) at any time t, if Q (0) = Qo. (c) Suppose that certain remains are discovered in which the current residual amount of carbon-14 is 20% of the original amount. Determine the age of these remains.
An important tool in archaeological research is the radiocarbon dating developed by the American chemist Willard F. Libby. This is the means of determining the age of certain wood and plant remains, and 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 years), measure 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 maount 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 Qo is the original amount, then the ration Q(t)/Qo can be determined, as long as this quantity is not too small. Present measurement techniques permite the use of this method for time periods of 50,000 years or more. (a) Assuming the Q satisfies the differential equation Q' = -rQ, determine the decay constant r for carbon-14 (hint: interpret the paragraph above). (b) Find an expression for Q(t) at any time t, if Q (0) = Qo. (c) Suppose that certain remains are discovered in which the current residual amount of carbon-14 is 20% of the original amount. Determine the age of these remains.
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Transcribed Image Text:An important tool in archaeological research is the radiocarbon dating developed by the American
chemist Willard F. Libby. This is the means of determining the age of certain wood and plant remains, and 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 years), measure 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 maount 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 Qo is the original amount, then the ration Q(t)/Qo can be determined,
as long as this quantity is not too small. Present measurement techniques permite the use of this method for time
periods of 50,000 years or more.
(a) Assuming the Q satisfies the differential equation Q' = -rQ, determine the decay constant r for carbon-14
(hint: interpret the paragraph above).
(b) Find an expression for Q(t) at any time t, if Q(0) = Qo.
(c) Suppose that certain remains are discovered in which the current residual amount of carbon-14 is 20% of the
original amount. Determine the age of these remains.
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