Question

(2 points) 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 stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5592 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' =-kC. k= (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 [1]: The New York Times, October 18, 1988.

Answer 1

Solution: Let, co = the amount of Sample remaining at any time t The clifferential equation is: . c'=-ke =) =kC all dt -> de - kot when to, co t=1 c =CE Integrating on both sides, -as face) je tam = ln (CF)-locco) = *t => in c.) = -Kt ->0 given Half -- iife = 5592 y2295 At Half time, final amount = xînitial amount - CE50:56

CE : 0.5 Glo when ce 20.5, t= t1 =5592 years From O, en(0.5) = -K(5592) ->> - Onco.5) 5592 (-0.6931471806) 5592 K = 0.0001239533 years 91 f = 0.91 on Substiteeting t= 40 days in 0; cloth contained 91. of original amount of cayhon je, ali = 0.91 From 0, to x ln (CE) 100 - Xeno.91 0.0001239533 - -| 0.0001239533 (-0.0943106 795) 761 . = 760 5565443599 Age = t = 761 years