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 5595 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.(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. How old is the Shroud of Turin, according to these data?

Answer 1

a)
`k = 0.000124`

b) According to these data, the Shroud of Turin has around 760 years.

Explanation:

The amount of carbon-14 is modeled by the following equation:

`C(t) = C_(0)e^(-kt)`

In which
`C_(0)` is the initial amount and k is the rate of decrease.

(a) Find the value of the constant k in the differential equation.

Half-life of 5595 years.

So
`C(5595) = 0.5C_(0)`

`C(t) = C_(0)e^(-kt)`

`0.5C_(0) = C_(0)e^(-5595k)`

`e^(-5595k) = 0.5`

Applying ln to both sides

`-5595k = -0.69`

`k = 0.000124`

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. How old is the Shroud of Turin, according to these data?

This is t when
`C(t) = 0.91C_(0)`

`C(t) = C_(0)e^(-kt)`

`0.91C_(0) = C_(0)e^(-0.000124t)`

`e^(-0.000124t) = 0.91`

Applying ln to both sides

`-0.000124t = -0.094`

`t = 760.57`

According to these data, the Shroud of Turin has around 760 years.