Question

the half - life of carbon - 14 is 5730 years. how long will it take for 7/8 of a sample of carbon - 14 to decay?
o 11,460 years
o 17,190 years
o 22,920 years
o 28,650 years

Explanation:

Step1: Determine remaining fraction

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

Step2: Relate remaining fraction to half - life formula

The formula for radioactive decay in terms of half - life $t_{1/2}$ 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 want $\frac{N}{N_0}=\frac{1}{8}=(\frac{1}{2})^n$. Solving for $n$, we find $n = 3$ since $(\frac{1}{2})^3=\frac{1}{8}$.

Step3: Calculate time passed

Given that the half - life $t_{1/2}=5730$ years and $n = 3$, the time $t$ passed is $t=n\times t_{1/2}$. So $t = 3\times5730=17190$ years.

Answer:

17,190 years