A piece of wood found in an ancient city has a carbon-14 to carbon-12 ratio that is one-eighth the carbon-14 to carbon-12 ratio of a tree growing nearby. How old is the piece of wood? (The half-life of carbon-14 is 5,715 years.)
k = 0.693/t1/2
Substitute k into the below equation.
ln(No/N) = kt
Use 100 for No
Use 12.5 for N (that's 1/8 of 100--or you could simply use 1/8 for that ratio).
k from above.
Solve for t in years.
2865 years
To determine the age of the piece of wood, we can use the concept of carbon dating. Carbon-14 (C-14) is a radioactive isotope of carbon that decays over time. By measuring the ratio of carbon-14 to carbon-12 (C-12) in a sample, we can estimate its age.
The half-life of carbon-14 is the time it takes for half of the C-14 atoms in a sample to decay. In this case, the half-life of carbon-14 is given as 5,715 years.
Now, let's break down the information given in the question:
1. The piece of wood found in the ancient city has a carbon-14 to carbon-12 ratio that is one-eighth the ratio of a nearby tree.
2. We are given the half-life of carbon-14.
To solve this, we need to understand the relationship between the ratios of carbon-14 and carbon-12 and the age of the sample.
Since the ratio of carbon-14 to carbon-12 in the wood is one-eighth of the ratio in the nearby tree, it means that the wood has undergone seven half-lives. (One-eighth is equal to 1/2^3, and 3 half-lives correspond to a reduction to 1/8th of the original amount.)
To calculate the number of years, we need to multiply the half-life of carbon-14 by the number of half-lives that have passed. In this case, it would be 7 half-lives multiplied by 5,715 years per half-life:
7 x 5,715 years = 40,005 years
Hence, the piece of wood is approximately 40,005 years old.