Digital SAT Math Practice Question #440

A radioactive isotope, Isotope A, decays over time according to the model \( A(t) = A_0 (0.5)^{\frac{t}{h}} \), where \( A_0 \) is the initial mass, \( A(t) \) is the mass after \( t \) days, and \( h \) is the half-life in days. A scientist observes that a sample of Isotope A decreases to exactly \( 12.5\% \) of its original mass in 24 days. A second radioactive isotope, Isotope B, decays such that its mass is modeled by the function \( B(t) = B_0 (0.25)^{\frac{t}{k}} \), where \( k \) is a positive constant. If both Isotope A and Isotope B have the exact same half-life, what is the value of \( k \)?