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

a) k = 0.00012491389

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

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 (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 = -(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 = -(ln(0.91))/(0.00012491389)`

`t = 755`

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