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?

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

So

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

Half-life of 5549 years.

This means that . We use this to find k. So

So

(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

So

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

d no solution

step-by-step explanation: