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 5549 years. Let C(t) be the amount of carbon-14 present 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 carbon-14 contained in freshly made cloth of the same material. How old was the Shroud of Turin at the time of this data?

Answer 1

Answer:

a) k = 0.00012491389

b) The Shroud of Turin was 755 years old at the time of this data.

Step-by-step explanation:

(a) Find the value of the constant k in the differential equation C' = -kC.

First we find the differential equation, by separation of variables. So

$\int \frac{C^{\prime}}{C} dt = -\int k dt$

So

$\ln{C} = -kt + K$

In which K is the constant of integration, representing the initial amount of substance. So

$C(t) = C(0)e^{-kt}$

Half-life of 5549 years.

This means that $C(5549) = 0.5C(0)$. We use this to find k. So

$C(t) = C(0)e^{-kt}$

$0.5C(0) = C(0)e^{-5549k}$

$e^{-5549k} = 0.5$

$\ln{e^{-5549k}} = \ln{0.5}$

$-5549k = \ln{0.5}$

$k = -\frac{\ln{0.5}}{5549}$

$k = 0.00012491389$

So

$C(t) = C(0)e^{-0.00012491389t}$

(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 carbon-14 contained in freshly made cloth of the same material. How old was the Shroud of Turin at the time of this data?

This is t for which $C(t) = 0.91C(0)$

So

$C(t) = C(0)e^{-0.00012491389t}$

$0.91C(0) = C(0)e^{-0.00012491389t}$

$e^{-0.00012491389t} = 0.91$

$\ln{e^{-0.00012491389t}} = \ln{0.91}$

$-0.00012491389t = \ln{0.91}$

$t = -\frac{\ln{0.91}}{0.00012491389}$

$t = 755$

The Shroud of Turin was 755 years old at the time of this data.