13. an isotopes half - life is 50 years. __ percent of a rock will be daughter isotope if the rock is 200…

Explanation:

Step1: Calculate number of half - lives

The number of half - lives $n$ is calculated by dividing the total time elapsed $t$ by the half - life $T_{1/2}$. Given $t = 200$ years and $T_{1/2}=50$ years, so $n=\frac{t}{T_{1/2}}=\frac{200}{50}=4$.

Step2: Calculate remaining parent isotope fraction

The fraction of the parent isotope remaining $N$ after $n$ half - lives is given by the formula $N = (\frac{1}{2})^n$. Substituting $n = 4$, we get $N=(\frac{1}{2})^4=\frac{1}{16}$.

Step3: Calculate daughter isotope fraction

The fraction of the daughter isotope is $1 - N$. So, $1-\frac{1}{16}=\frac{16 - 1}{16}=\frac{15}{16}$. To convert this to a percentage, we multiply by 100: $\frac{15}{16}\times100=\frac{1500}{16}=93.75%$.

Answer:

93.75%