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Radiocarbon dating: Carbon-14 is a radioactive isotope of
Chapter 4, Problem 20E(choose chapter or problem)
Radiocarbon dating: Carbon- 14 is a radioactive isotope of carbon that decays by emitting a beta particle. In the earth's atmosphere, approximately one carbon atom in \(10^{12}\) is carbon-14. Living organisms exchange carbon with the atmosphere, so this same ratio holds for living tissue. After an organism dies, it stops exchanging carbon with its environment, and its carbon-14 ratio decreases exponentially with time. The rate at which beta particles are emitted from a given mass of carbon is proportional to the carbon- 14 ratio, so this rate decreases exponentially with time as well. By measuring the rate of beta emissions in a sample of tissue, the time since the death of the organism can be estimated. Specifically, it is known that years after death, the number of beta particle emissions occurring in any given time interval from 1g of carbon follows a Poisson distribution with rate \(\lambda=15.3 e^{-0.0001210 t}\) events per minute. The number of years
since the death of an organism can therefore be expressed in terms of :
$$t=\frac{\ln 15.3-\ln \lambda}{0.0001210}$$
An archaeologist finds a small piece of charcoal from an ancient campsite. The charcoal contains \(1 \mathrm{~g}\) of carbon.
a. Unknown to the archaeologist, the charcoal is 11,000 years old. What is the true value of the emission rate?
b. The archaeologist plans to count the number X of emissions in a 25 minute interval. Find the mean and standard deviation of X.
c. The archaeologist then plans to estimate \(\lambda\) with \(\hat{\lambda}=X / 25\). What is the mean and standard deviation of \(\hat{\lambda}\)?
d. What value for \(\hat{\lambda}\) would result in an age estimate of 10,000 years?
e. What value for \(\hat{\lambda}\) would result in an age estimate of 12,000 years?
f. What is the probability that the age estimate is correct to within \(\pm 1000\) years?
Equation Transcription:
Text Transcription:
10^12
Tilde = 15.3e^-0.0001210t
t = ln15.3-ln tilde/0.0001210
Lambda
Lambda hat = X/25
Lambda hat
Plus or minus 1000
Questions & Answers
QUESTION:
Radiocarbon dating: Carbon- 14 is a radioactive isotope of carbon that decays by emitting a beta particle. In the earth's atmosphere, approximately one carbon atom in \(10^{12}\) is carbon-14. Living organisms exchange carbon with the atmosphere, so this same ratio holds for living tissue. After an organism dies, it stops exchanging carbon with its environment, and its carbon-14 ratio decreases exponentially with time. The rate at which beta particles are emitted from a given mass of carbon is proportional to the carbon- 14 ratio, so this rate decreases exponentially with time as well. By measuring the rate of beta emissions in a sample of tissue, the time since the death of the organism can be estimated. Specifically, it is known that years after death, the number of beta particle emissions occurring in any given time interval from 1g of carbon follows a Poisson distribution with rate \(\lambda=15.3 e^{-0.0001210 t}\) events per minute. The number of years
since the death of an organism can therefore be expressed in terms of :
$$t=\frac{\ln 15.3-\ln \lambda}{0.0001210}$$
An archaeologist finds a small piece of charcoal from an ancient campsite. The charcoal contains \(1 \mathrm{~g}\) of carbon.
a. Unknown to the archaeologist, the charcoal is 11,000 years old. What is the true value of the emission rate?
b. The archaeologist plans to count the number X of emissions in a 25 minute interval. Find the mean and standard deviation of X.
c. The archaeologist then plans to estimate \(\lambda\) with \(\hat{\lambda}=X / 25\). What is the mean and standard deviation of \(\hat{\lambda}\)?
d. What value for \(\hat{\lambda}\) would result in an age estimate of 10,000 years?
e. What value for \(\hat{\lambda}\) would result in an age estimate of 12,000 years?
f. What is the probability that the age estimate is correct to within \(\pm 1000\) years?
Equation Transcription:
Text Transcription:
10^12
Tilde = 15.3e^-0.0001210t
t = ln15.3-ln tilde/0.0001210
Lambda
Lambda hat = X/25
Lambda hat
Plus or minus 1000
ANSWER:
Solution
Step 1 of 6
a) we have to find the true value of the emission rate when the charcoal is 11000 years old
Given
Here t=11000 years
=168279.6