Final answer:
To calculate the time it takes for a radioactive isotope to reach 3.125% of its original amount, we determine the number of half-lives required. For Isotope Y with an 88-minute half-life, it takes 5 half-lives (440 minutes) to reach 3.125% radioactivity.
Step-by-step explanation:
The problem you're asking about involves calculating how long it takes for a radioactive isotope to decrease to a certain percentage of its original amount, specifically using half-life calculations. In this case, Isotope Y has a half-life of 88 minutes, and you want to know how long it will take for the radioactive isotope to end up with 3.125% remaining. With each half-life, the amount of radioactive material decreases by half. To reach 3.125%, which is 1/32 of the original amount, we need to halve the original amount five times (25 = 32).
Doing the math: 88 minutes/half-life × 5 half-lives = 440 minutes.
Therefore, it will take 440 minutes for the sample of Isotope Y to be just 3.125% radioactive.