calculus
posted by j on .
Carbon14 is a radioactive substance produced in the Earth's atmosphere and then absorbed by plants and animals on the surface of the earth. It has a halflife (the time it takes for half the amount of a sample to decay) of approximately 5730 years. Using this known piece of information, scientists can date objects such as the Dead Sea Scrolls. The function N = N0eλt represents the exponential decay of a radioactive substance. N is the amount remaining after time t in years, N0 is the initial amount of the substance and λ is the decay constant.
1. Find the rate of change of an initial amount of 1 gm of carbon14 found in the scrolls, if the decay constant is given as λ = 1.21 x 104.
2. If the percentage of carbon14 atoms remaining in a sample is 79%, how old is the sample?

1.
dN/dt = λN
so the initial rate of change
= 1.21*10^4 × 1 g/year
= 0.000121 g/year
2.
N = N_{0}e^{λt}
eλt = N/N_{0}
taking logs on both sides
λt = ln(N/N_{0})
t = ln(N/N_{0})/λ (before present, year 1950)
= ln(0.79)/(1.21*10^{4}) (BP)
= 1948 (BP)