Using the concept of half-life, the total visit time to safely pass through the radiation detector without setting off the alarm is calculated to be 90 minutes.
Step-by-step explanation:
The amount of radioactive dye decreases over time, and we need to determine when it will fall below the threshold of 2 milligrams to safely pass through a radiation detector. Given that 14 milligrams of dye were initially administered, and after 20 minutes the amount reduces to 8 milligrams, we can infer that this is a half-life problem where the half-life is 20 minutes (since the dye amount halved from 14 to 8).
We then use the half-life formula to determine the total time it takes for the dye to decay to less than 2 milligrams from the initial 14 milligrams:
Initial amount: 14 mg
Amount after 20 minutes: 8 mg (half-life)
Amount after 40 minutes: 4 mg (second half-life)
Amount after 60 minutes: 2 mg (third half-life)
With each half-life being 20 minutes, the dye amount halves every 20 minutes. Thus, it takes 60 minutes for the dye to reach 2 milligrams. However, the detector will sound the alarm if more than 2 milligrams are present. We know that the dye amount at 60 minutes is exactly 2 mg, which is the limit. Waiting half of another half-life (10 minutes), the dye amount will halve to 1 mg, which is safely below the alarm threshold.
Therefore, the total visit time is the time taken for the initial 20 minutes, plus 60 minutes for the dye to halve three times, plus an additional 10 minutes to ensure the dye amount is safely below 2 milligrams, which totals to 90 minutes.