Understanding Radiocarbon Activity Decrease

The question states that the radiocarbon activity in a sample is now 1/4 of its initial activity. This means the amount of radioactive carbon has reduced significantly over time.

We are given the half-life of radiocarbon ($T_{1/2}$) as 5700 years. The half-life is the time it takes for half of the radioactive substance to decay.

Calculating Sample Age from Activity

Since the activity is 1/4 of the initial value, we can determine how many half-lives have passed:

• After 1 half-life: Activity = 1/2 of initial.
• After 2 half-lives: Activity = 1/2 of (1/2) = 1/4 of initial.

Therefore, 2 half-lives have passed for the activity to become 1/4 of the initial amount.

Determining the Total Age

To find the total age of the sample, we multiply the number of half-lives by the duration of one half-life:

Age ($t$) = Number of half-lives $\times$ Half-life duration ($T_{1/2}$)

Age ($t$) = $2 \times 5700 \text{ years}$

Age ($t$) = $11400 \text{ years}$

The age of the sample is 11400 years.