Question

Carbon from a cypress beam obtained from the tomb of an ancient Egyptian king gave 9.2 disintegrations/minute of C-14 per gram of carbon. Carbon from living material gives 15.3 disintegrations/min of C-14 per gram of carbon. Carbon-14 has a half-life of 5730 years. How old is the cypress beam?

Short answer

Question: Using the given C-14 disintegration rates of a cypress beam from an ancient Egyptian tomb (9.2 disintegrations/min per gram of carbon) and a sample of living material (15.3 disintegrations/min per gram of carbon), along with the half-life of Carbon-14 (5730 years), determine the age of the cypress beam. Answer: The cypress beam is approximately 2693 years old.

Step-by-step solution

Identify the given information

We are given the following information: 1. Disintegration rate of cypress beam: 9.2 disintegrations/min per gram of carbon 2. Disintegration rate of living material: 15.3 disintegrations/min per gram of carbon 3. Half-life of Carbon-14: 5730 years Let's denote the disintegration rate of the cypress beam as R_beam, the disintegration rate of living material as R_living, and the half-life as t_1/2.

Write the exponential decay formula

The formula for exponential decay is as follows: N(t) = N_0 * (1/2)^(t/t_1/2), where: - N(t) is the amount of the substance at time t, - N_0 is the initial amount of the substance, - t is the time that has elapsed, and - t_1/2 is the half-life of the substance. Since we are given the disintegration rates, we can use them to find the ratio of the amount of carbon in the cypress beam to the amount in living material. We have: R_beam = N(t) * k R_living = N_0 * k where k is a constant.

Write the equation for the ratio of amounts of carbon

Now we can write the equation for the ratio of the amount of carbon in the cypress beam to that in the living material: R_beam/R_living = N(t)/N_0 = (1/2)^(t/t_1/2)

Substitute the given values into the equation

Now we substitute the given values into the equation and solve for the age t: (9.2/15.3) = (1/2)^(t/5730)

Solve for the age t

To solve for t, we first take the natural logarithm of both sides of the equation: ln(9.2/15.3) = ln((1/2)^(t/5730)) Now we apply the logarithmic property ln(a^b) = b*ln(a): ln(9.2/15.3) = (t/5730) * ln(1/2) We can now solve for t: t = (5730 * ln(9.2/15.3)) / ln(1/2)

Calculate the age of the cypress beam

Finally, we can calculate the age of the cypress beam using the previous expression: t ≈ (5730 * ln(9.2/15.3)) / ln(1/2) ≈ 2692.98 So the cypress beam is approximately 2693 years old.