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 5566 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 Cr KC. (b) In 1988 three teams of scientists found that the Shroud ofTurin, 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[ij. How old is the Shroud of Turin, according to these data? Age years [11: The New York Times, October 18, 1988.

Answer 1

Answer:

a) k = 0.0001245 /year

b) The Shroud of Turin is 757.52 years old.

Step-by-step explanation:

a) dC/dt = - KC

dC/C = - kdt

Integrating the left hand side from C₀ to C₀/2 and the right hand side from 0 to t(1/2) (where t(1/2) is the radioactive isotope's half life)

In [(C₀/2)/C₀] = - k t(1/2)

In (1/2) = - k t(1/2)

- In 2 = - k t(1/2)

k = (In 2)/t₍₁,₂₎

t(1/2) is given in the question to be 5566 years

k = (In 2)/5566 = 0.0001245 /year

b) dC/C = - kdt

Integrating the left hand side from C₀ to C and the right hand side from 0 to t

In (C/C₀) = - kt

C/C₀ = e⁻ᵏᵗ

C = C₀ e⁻ᵏᵗ

When C = 91% of C₀ = 0.91 C₀, solve for t

0.91 C₀ = C₀ e⁻ᵏᵗ

0.91 = e⁻ᵏᵗ

e⁻ᵏᵗ = 0.91

- kt = In 0.91 = -0.00943

t = - 0.00943/(-0.0001245) = 757.52 years.

Answer 2

Using an exponential function, it is found that:

a) k = 0.00012453237

b) According to these data, the Shroud of Turin is 757 years old.

The exponential function for the amount of a decaying substance is given by:

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

In which:

• A(0) is the initial amount.
• k is the decay rate, as a decimal.

Item a:

The half-life is of 5566 years, hence $A(5566) = 0.5A(0)$, and this is used to find k.

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

$0.5A(0) = A(0)e^{-5566k}$

$e^{-5566k} = 0.5$

$\ln{e^{-5566k}} = \ln{0.5}$

$-5566k = \ln{0.5}$

$k = -\frac{\ln{0.5}}{5566}$

$k = 0.00012453237$

Item b:

The time is t for which A(t) = 0.91A(0), hence:

$A(t) = A(0)e^{-0.00012453237t}$

$0.91A(0) = A(0)e^{-0.00012453237t}$

$e^{-0.00012453237t} = 0.91$

$\ln{e^{-0.00012453237t}} = \ln{0.91}$

$-0.00012453237t = \ln{0.91}$

$t = -\frac{\ln{0.91}}{0.00012453237}$

$t = 757$

According to these data, the Shroud of Turin is 757 years old.

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