Carbon-14 Dating Problem: Carbon-14 is an isotope of carbon that forms when radiation

Chapter 7, Problem 6

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Carbon-14 Dating Problem: Carbon-14 is an isotope of carbon that forms when radiation from the Sun strikes ordinary carbon dioxide in the atmosphere. Thus, plants such as trees, which get their carbon dioxide from the atmosphere, contain small amounts of carbon-14. Once a particular part of a plant stops growing, no more new carbon-14 is absorbed by that part. The carbon-14 in that part decays slowly, transmuting into nitrogen-14. Let P be the percentage of carbon-14 that remains in a part of a tree that grew a number of years ago, t. a. The instantaneous rate of change of P with respect to t is directly proportional to P. Use this fact to write a differential equation that relates these two variables. b. Solve the differential equation for P in terms of t. Use the fact that the half-life of carbon-14 is 5750 yr. That is, if P = 100 when t = 0, then P = 50 when t = 5750. c. The oldest living trees in the world are the bristlecone pines in the White Mountains of California. Scientists have counted 4000 growth rings in the trunk of one of these trees, meaning that the innermost ring grew 4000 years ago. What percentage of the original carbon-14 would you expect to find remaining in this innermost ring? d. A piece of wood claimed to have come from Noahs Ark is found to have 48.37% of itscarbon-14 remaining. It has been suggestedthat the Great Flood occurred in 4004 B.C.E.Is the wood old enough to come from NoahsArk? Explain.e. Plot the graph of P versus t for times from0 years through at least 20,000 years. Use yourgraphers TRACE feature to demonstrate that your answers to parts ce are correct by showingthat they lie on this graph. Sketch the results.