Answer:
To determine the age of the ancient cloth, we can use the concept of radioactive decay and the half-life of Carbon-14.
Carbon-14 is a radioactive isotope of carbon, which decays over time into nitrogen-14 through beta decay. The half-life of Carbon-14 is 5614 years, which means that after 5614 years, half of the original amount of Carbon-14 in a sample will have decayed.
In this case, the cloth contains only ¼ of its original Carbon-14. This means that three half-lives have passed since the cloth was first created, as each half-life reduces the amount of Carbon-14 by half.
To determine the age of the cloth, we can use the following formula:
N = N0(1/2)^t/T
where N is the current amount of Carbon-14 in the cloth, N0 is the original amount of Carbon-14 in the cloth, t is the time that has passed, and T is the half-life of Carbon-14.
We know that N = ¼ N0, and T = 5614 years. Plugging these values into the formula, we get:
¼ N0 = N0(1/2)^(3/T)
Solving for t, we get:
t = (3/T) * log(2)
Substituting in T = 5614 years, we get:
t = (3/5614) * log(2) ≈ 1,684 years
Therefore, the cloth is approximately 1,684 years old.
In summary, we can use the concept of radioactive decay and the half-life of Carbon-14 to determine the age of the ancient cloth. By knowing the current amount of Carbon-14 in the cloth, we can calculate the time that has passed since it was first created using a simple formula. In this case, the cloth is approximately 1,684 years old.