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Previous Problem Problem List Next Problem (1 point) The radioactive isotope carbon-14 is present in small quantitie in all ite forms, and it is çonstantly replenished until the organism dies,after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5543 years. Suppose C(t) is the amount of carbon-14 present at time t. (a) Find the value of the constant k in the differential equation C, =-k C. b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 contained in freshly made cloth of the same material[1]. How was old the Shroud of Turin in 1988, according to these data? Age = years : The New York Times, October 18, 1988. Note: You can earn pärtial credit on this problem.

Explanation / Answer

a) C" = -kC dC/dt = -kC Separate the variables or Separable variable dC/C = -k.dt Apply Integration ln(C) = -kt + c1 C = e^(-kt+c1) C = A.e^(-kt), where A = e^c1 At t = 0, C = A Half life is 5543 years At t = 5543, C = A/2 A/2 = A.e^(-5543k) 1/2 = e^(-5543k) -5543k = ln(1/2) 5543k = ln(2) k = 0.00013 (approximately) b) C = C(0).(1/2)^(t/5543) C/C(0) = (1/2)^(t/5543) 0.91 = (1/2)^(t/5730) (t/5543)ln(1/2) = ln(0.91) t/5543 = ln(0.91)/ln(1/2) t = 754.2 years

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