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 1012 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 t years after death, the number of beta particle emissions occurring in any given time interval from 1 g of carbon follows a Poisson distribution with rate i = 15.3e-0.0001210:events per minute. The number of yearst since the death of an organism can therefore be expressed in terms of 2: In 15.3 – In i t = 0.0001210 An archaeologist finds a small piece of charcoal from an ancient campsite. The charcoal contains 1 g of carbon. Unknown to the archaeologist, the charcoal is 11,000 years old. What is the true value of the emission rate 1? a. b. The archaeologist plans to count the number X of emissions in a 25 minute interval. Find the mean and standard deviation of X. The archaeologist then plans to estimate à with î=x/25 · What is the mean and standard deviation of ? C. What value for î would result in an age estimate of 10,000 years? d. What value for î would result in an age estimate of 12,000 years? e. f. What is the probability that the age estimate is correct to within ±1000 years?
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 1012 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 t years after death, the number of beta particle emissions occurring in any given time interval from 1 g of carbon follows a Poisson distribution with rate i = 15.3e-0.0001210:events per minute. The number of yearst since the death of an organism can therefore be expressed in terms of 2: In 15.3 – In i t = 0.0001210 An archaeologist finds a small piece of charcoal from an ancient campsite. The charcoal contains 1 g of carbon. Unknown to the archaeologist, the charcoal is 11,000 years old. What is the true value of the emission rate 1? a. b. The archaeologist plans to count the number X of emissions in a 25 minute interval. Find the mean and standard deviation of X. The archaeologist then plans to estimate à with î=x/25 · What is the mean and standard deviation of ? C. What value for î would result in an age estimate of 10,000 years? d. What value for î would result in an age estimate of 12,000 years? e. f. What is the probability that the age estimate is correct to within ±1000 years?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
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Transcribed Image Text: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 1012 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 t years after death, the number of beta particle emissions occurring in any
given time interval from 1 g of carbon follows a Poisson distribution with rate
i = 15.3e-0.0001210:events per minute. The number of yearst since the death of an organism
can therefore be expressed in terms of 2:
In 15.3 – In i
t =
0.0001210
An archaeologist finds a small piece of charcoal from an ancient campsite. The charcoal
contains 1 g of carbon.
Unknown to the archaeologist, the charcoal is 11,000 years old. What is the true value
of the emission rate 1?
a.
b.
The archaeologist plans to count the number X of emissions in a 25 minute interval.
Find the mean and standard deviation of X.
The archaeologist then plans to estimate à with î=x/25 · What is the mean and
standard deviation of ?
C.
What value for î would result in an age estimate of 10,000 years?
d.
What value for î would result in an age estimate of 12,000 years?
e.
f.
What is the probability that the age estimate is correct to within ±1000 years?
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