The radioactive isotope carbon-14 is present in small quantities in all life for

Question

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 5555 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

Explanation / Answer

decay y=e^(-kt)
growth y=e^(+kt)

you have decay so

C=e^(-kt) f

C'=-k*e^(-kt)=-k*C also fits.

now to find k:
1/2=e^(-k*5555)

ln(1/2)=-k*5555
-ln(2)=-k*5555
k=ln(2)/5555

0.91=e^(-ln(2)*t/5555)
ln(0.91)=-ln(2)*t/5555
t=(-ln(0.91)*5555)/ln(2) = 755.82 = 756 years

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