Carbon Dating the Shroud of TurinDate: 05/16/2000 at 21:45:08 From: Catherine Sullivan Subject: radioactive decay Please help me with the following: The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to carbon-12 at a rate proportional to the amount of C-14 present, with a half-life of 5730 years. Suppose C(t) is the amount of C-14 at time t. a) Find the value of the constant k in the differential equation: C" = -kC b) In 1988, three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of C-14 contained in freshly made cloth of the same material. How old is the Shroud according to the data? Note: for part a, I know how to solve for k using the equation C = C*e^(kt) but I need to do it from the differential equation. I took the integral of both sides but was left with a first derivative and too many unknowns. Please help! Thanks Date: 05/17/2000 at 07:56:48 From: Doctor Anthony Subject: Re: radioactive decay >a) Find the value of the constant k in the differential equation: > > C" = -kC I think you mean: C' = -kC dC/dt = -kC dC/C = -k.dt ln(C) = -kt + constant C = e^(-kt+constant) C = A.e^(-kt) where A = e^constant at t = 0, C = A at t = 5730, C = A/2 A/2 = A.e^(-5730k) 1/2 = e^(5730k) -5730k = ln(1/2) k = 1.2097 x 10^(-4) >b) In 1988, three teams of scientists found that the Shroud of Turin, >which was reputed to be the burial cloth of Jesus, contained 91% of >the amount of C-14 contained in freshly made cloth of the same >material. How old is the Shroud according to the data? It is not necessary to use the formula derived above. We can say: C = C(0).(1/2)^(t/5730) C/C(0) = (1/2)^(t/5730) 0.91 = (1/2)^(t/5730) taking logs: (t/5730)ln(1/2) = ln(0.91) t/5730 = ln(0.91)/ln(1/2) = 0.13606 t = 779.6 years So the shroud is about 780 years old. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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