calculating the time required for a sample to decay\nthe half - life of carbon - 14 is 5730 years. how long…

Explanation:

Step1: Determine remaining fraction

If 7/8 has decayed, the remaining fraction of carbon - 14 is $1-\frac{7}{8}=\frac{1}{8}$.

Step2: Use half - life formula

The formula for radioactive decay is $N = N_0(\frac{1}{2})^n$, where $N$ is the final amount, $N_0$ is the initial amount, and $n$ is the number of half - lives. Since $\frac{N}{N_0}=\frac{1}{8}=(\frac{1}{2})^3$, $n = 3$.

Step3: Calculate time

Each half - life is 5730 years. So the time $t= n\times$ half - life. Substituting $n = 3$ and half - life = 5730 years, we get $t=3\times5730 = 17190$ years.

Answer:

17,190 years