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 5545 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 old is the Shroud of Turin, according to these data?
Age = ??? years
(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 old is the Shroud of Turin, according to these data?
Age = ??? years
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decay y=e^(-kt)
growth y=e^(+kt)
you have decay so
C=e^(-kt) fits.
C'=-k*e^(-kt)=-k*C also fits.
now to find k:
1/2=e^(-k*5545)
ln(1/2)=-k*5545
-ln(2)=-k*5545
k=ln(2)/5545
0.91=e^(-ln(2)*t/5545)
ln(0.91)=-ln(2)*t/5545
t=(-ln(0.91)*5545/ln(2)
growth y=e^(+kt)
you have decay so
C=e^(-kt) fits.
C'=-k*e^(-kt)=-k*C also fits.
now to find k:
1/2=e^(-k*5545)
ln(1/2)=-k*5545
-ln(2)=-k*5545
k=ln(2)/5545
0.91=e^(-ln(2)*t/5545)
ln(0.91)=-ln(2)*t/5545
t=(-ln(0.91)*5545/ln(2)