Find the value of the constant k in the differential equation C?=?kC. The radioa

Question

Find the value of the constant k in the differential equation C?=?kC.

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 5570 years. Suppose C(t) is the amount of carbon-14 present at time t. Find the value of the constant k in the differential equation C' = -kC. 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 old is the Shroud of Turin, according to these data?

Explanation / Answer

a)

given C'=-kC

dC/dt =-kC

dC/C=-kdt

integrate on both sides

dC/C= -kdt

lnC=-kt +c

C=e-kt+c

C=e-ktec

C=pe-kt where p is constant

at t=0

C0=pe-0

C0=p where Co is initial amount

so decay equation is C=Coe-kt

given half life is 5570 years => at 5570 years C=Co/2

Co/2=Coe-k5570

e-k5570=1/2

-5570k =ln(1/2)

k=-(ln(1/2))/5570

k=0.00012444 per year

C=Coe-0.00012444t

b)contains 91% of carbon -14 of the original material

=>C=0.91Co

0.91Co=Coe-0.00012444t

e-0.00012444t=0.91

-0.00012444t=ln(0.91)

t=-(ln(0.91))/(0.00012444)

t =758 years

age of cloth is 758 years

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