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A1 - Applied Math 250 Assignment#1 Spring 2009 Note 4066 A...

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Unformatted text preview: Applied Math 250 Assignment #1 Spring 2009 Note: 4066. A/ B/ due May 19th Assignments are due by noon on the due date, in the drop boxes across from MC Problem Set 1 #li, ii, #2i,iv, part (a) of #2viii, #3, #41 Other strongly recommended problems (not for submission): #lv, #2vi, part (a) of #2111, vii, #4111 Hint for #1: The constant of integration should not appear in the DE. You’ll need to eliminate it. Eg, in #liii differentiation gives 2% = 1 —— Ke‘”, but we know K = yej—f, so i1 = 1 — y — :13. Radiocarbon Dating. An important tool in urcmoloand research is radiocarbon dating. This is a means of determining the age of certain wood and plant remains, hence of animal or human bones or artifacts found buried at the same levels. The procedure was deveIOped by the American chemist Willard Libby (1908-1980) in the early 19503 and resulted in his winning the Nobel prize for chemistry in 1960. 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-His long (approximately 5568 years), measureable amounts of carbonsléi remain after many thousands of years. Libby showed that even if a tiny fraction of the orig- inal‘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 Q0 is the original amount, then the ratio Q(t)/Qo can be determined, at least if this quantity is not too small. Present measurement techniques permit the use of this method for time periods up to about 100,000 years, after which the amount of carbon- 14 remaining is only about 4 x 10‘6 of the original amount. To model the phenomenon of radioactive decay, it is assumed that the rate dQ/dt at which carbon-l4 decays is proportional to the amount (more precisely, the number of nuclei) Q(t) of carbon-14, with proportionality constant 1:, remaining at time t. a) Expressthis relationship as a DB for Q(t), but set it up in such a way that It > 0 (you ' may need to insert a minus sign somewhere—think carefully about whether you have growth or decay). b) Using the DE, sketch typical solutions as you did in # 2. c) Solve the DE for Q(t), and find the solution satisfying Q(O) = Q0. Note that k is still an unknown constant. (:1) Given the half-life of carbon-l4 (above), find k. e) Suppose that certain remains are discovered in which the current residual amount of carbon-14 is 2 percent of the original amount. Determine the age of these remains. ...
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