the half - life of carbon - 14 is 5730 years. how long will it take for 7/8 of a sample of carbon - 14 to…

Explanation:

Step1: Determine remaining fraction

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

Step2: Relate remaining fraction to 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. We know $\frac{N}{N_0}=\frac{1}{8}=(\frac{1}{2})^n$. Solving for $n$, we find $n = 3$ since $(\frac{1}{2})^3=\frac{1}{8}$.

Step3: Calculate time passed

Since each half - life is 5730 years and $n = 3$, the time $t$ passed is $t=5730\times n$. Substituting $n = 3$ into the formula, we get $t = 5730\times3=17190$ years.

Answer:

17,190 years