Question 3.5.2: Carbon-14 Dating: The Dead Sea Scrolls a. Use the fact that ...
Carbon-14 Dating: The Dead Sea Scrolls
a. Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14.
b. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls.
a. We begin with the exponential decay model A=A_0 e^{k t}. We know that k < 0 because the problem involves the decay of carbon-14. After 5715 years (t = 5715), the amount of carbon-14 present, A, is half the original amount, A_0. Thus, we can substitute \frac{A_0}{2} for A in the exponential decay model. This will enable us to find k, the decay rate.
A=A_0 e^{k t} Begin with the exponential decay model.
\frac{A_0}{2}=A_0 e^{k 5715} After 5715 years (t = 5715), A=\frac{A_0}{2} (because the amount present, A is half the original amount, A_0).
\frac{1}{2}=e^{5715 k} Divide both sides of the equation by A_0.
\ln \left(\frac{1}{2}\right)=\ln e^{5715 k} Take the natural logarithm on both sides.
\ln \left(\frac{1}{2}\right)=5715 k Simplify the right side using \ln e^x=x.
k=\frac{\ln \left(\frac{1}{2}\right)}{5715} \approx-0.000121 Divide both sides by 5715 and solve for k.
Substituting -0.000121 for k in the decay model, A=A_0 e^{k t}, the model for carbon- 14 is
A=A_0 e^{-0.000121 t}.
b. In 1947, the Dead Sea Scrolls contained 76% of their original carbon-14. To find their age in 1947, substitute 0.76 A_0 for A in the model from part (a) and solve for t.
A=A_0 e^{-0.000121 t} This is the decay model for carbon-14.
0.76 A_0=A_0 e^{-0.000121 t} A, the amount present, is 76% of the original amount, so A=0.76 A_0.
0.76=e^{-0.000121 t} Divide both sides of the equation by A_0.
\ln 0.76=\ln e^{-0.000121 t} Take the natural logarithm on both sides.
ln 0.76 = –0.000121t Simplify the right side using \ln e^x=x.
t=\frac{\ln 0.76}{-0.000121} \approx 2268 Divide both sides by –0.000121 and solve for t.
The Dead Sea Scrolls are approximately 2268 years old plus the number of years between 1947 and the current year.