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9. An important tool in archeological research is radiocarbon dating, developed by the American chemist Willard F. Libby. This is a 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), 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 Qo is the original amount, then the ratio Q(t)/Qo can be determined, as long as this quantity is not too small. Present measurement techniques permit the use of this method for time periods of 50,000 years or more. a. Assuming that satisfies the differential equation Q' = -rQ, determine the decay constant r for carbon-14. b. Find an expression for Q(t) at any time t, if Q(0) c. Suppose that certain remains are discovered in which the lo. current residual amount of carbon-14 is 20% of the original amount. Determine the age of these remains. =

9. An important tool in archeological research is radiocarbon dating, developed by the American chemist Willard F. Libby. This is a 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), 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 Qo is the original amount, then the ratio Q(t)/Qo can be determined, as long as this quantity is not too small. Present measurement techniques permit the use of this method for time periods of 50,000 years or more. a. Assuming that satisfies the differential equation Q' = -rQ, determine the decay constant r for carbon-14. b. Find an expression for Q(t) at any time t, if Q(0) c. Suppose that certain remains are discovered in which the lo. current residual amount of carbon-14 is 20% of the original amount. Determine the age of these remains. =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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rate of 6% to purchase a condominium. Anticipating steady increases, the buyer expects to make payments at a monthly rate of 800+10t, where t is the number of months since the loan was made. a. Assuming that this payment schedule can be maintained, when will the loan be fully paid? b. Assuming the same payment schedule, how large a loan could be paid off in exactly 20 years? 4 9. An important tool in archeological research is radiocarbon dating, developed by the American chemist Willard F. Libby. This is a 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), 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 Qo is the original amount, then the ratio Q(t)/Qo can be determined, as long as this quantity is not too small. Present measurement techniques permit the use of this method for time periods of 50,000 years or more. a. Assuming that satisfies the differential equation Q' = -rQ, determine the decay constant r for carbon-14. 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. N 10. Suppose that a certain population has a growth rate that equation varies with time and that this population satisfies the differential dy dt = (0.5+ sint) aj 5 a. If y(0) 1, find (or estimate) the time at which the population has doubled. Choose other initial conditions and population. determine whether the doubling time T depends on the initial b. Suppose that the growth rate is replaced by its aver 1/10. Determine the doubling tin b. рс C. ea G ve 12. N changes and that coffee o of 200°E a room a 150°F. 13. 13. He based on equation where u the absol depending is much approxim Suppose t a medium a. D equat G b. N the a equat no mo N 14. C internal tem satisfi
Transcribed Image Text:rate of 6% to purchase a condominium. Anticipating steady increases, the buyer expects to make payments at a monthly rate of 800+10t, where t is the number of months since the loan was made. a. Assuming that this payment schedule can be maintained, when will the loan be fully paid? b. Assuming the same payment schedule, how large a loan could be paid off in exactly 20 years? 4 9. An important tool in archeological research is radiocarbon dating, developed by the American chemist Willard F. Libby. This is a 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), 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 Qo is the original amount, then the ratio Q(t)/Qo can be determined, as long as this quantity is not too small. Present measurement techniques permit the use of this method for time periods of 50,000 years or more. a. Assuming that satisfies the differential equation Q' = -rQ, determine the decay constant r for carbon-14. 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. N 10. Suppose that a certain population has a growth rate that equation varies with time and that this population satisfies the differential dy dt = (0.5+ sint) aj 5 a. If y(0) 1, find (or estimate) the time at which the population has doubled. Choose other initial conditions and population. determine whether the doubling time T depends on the initial b. Suppose that the growth rate is replaced by its aver 1/10. Determine the doubling tin b. рс C. ea G ve 12. N changes and that coffee o of 200°E a room a 150°F. 13. 13. He based on equation where u the absol depending is much approxim Suppose t a medium a. D equat G b. N the a equat no mo N 14. C internal tem satisfi
Expert Solution
Step 1

Given,

It is believed that the mass of carbon-14 diminishes at a rate that is proportionate to its current concentration,

dQdt=-Q

By multiplying the constant r to the right side, this proportionality can be converted into an equation,

Q'=-rQ

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