To determine the age of an object based on the remaining percentage of a radioactive element, we can use the concept of half-lives.
The half-life of the radioactive element is given as 800 years. If 12.5% of the radioactive atoms remain, we can determine how many half-lives have passed.
12.5% can be expressed as a fraction:
\[ 12.5% = \frac{12.5}{100} = \frac{1}{8} \]
Now we can relate this fraction to half-lives. Each half-life reduces the quantity of the radioactive element by half. So:
- After 1 half-life (800 years): \( \frac{1}{2} \)
- After 2 half-lives (1600 years): \( \frac{1}{4} \)
- After 3 half-lives (2400 years): \( \frac{1}{8} \)
Since we have \( \frac{1}{8} \) remaining, we can conclude that 3 half-lives have elapsed.
To find the total time that has passed:
\[ \text{Total time} = 3 \times \text{Half-life} = 3 \times 800 \text{ years} = 2400 \text{ years} \]
Therefore, the age of the object is 2400 years.