Choose the one alternative that best completes the statement or answers the question.
-A certain radioactive isotope has a half-life of approximately 1250 years. How many years to the nearest year would be required for a given amount of this isotope to decay to $70 \%$ of that amount?
A) $375 \mathrm { yr }$
B) $2171 \mathrm { yr }$
C) $573 \mathrm { yr }$
D) $643 \mathrm { yr }$

Question

Quizplus · Accepted Answer

The decay of a radioactive isotope can be modeled by the formula $N = N_0(1/2)^{t/T}$, where $N$ is the final amount, $N_0$ is the initial amount, $t$ is the time in years, and $T$ is the half-life of the isotope. To find the time required for the isotope to decay to 70% of its original amount, we set $N/N_0 = 0.7$ and solve for $t$. Substituting $T = 1250$ years, we get $0.7 = (1/2)^{t/1250}$. Solving for $t$, we find $t \approx 643$ years.

Choose the One Alternative That Best Completes the Statement or Answers