Radon-222 exponential decay from 450 mg — daily factor 0.83 (half-life ≈ 3.72 days)

We are given the model m(t) = 450(0.83)^t, where m(t) is the mass in milligrams after t days.

1. Initial amount: At t = 0, m(0) = 450(0.83)^0 = 450 mg. That is the starting mass.
2. Daily change (what 0.83 means): The factor 0.83 is the amount remaining after one day. Multiplying by 0.83 each day means the sample keeps 83% of its mass from the previous day and loses 17% per day (100% − 83% = 17%).
3. Examples:
   • After 1 day: m(1) = 450(0.83) = 373.5 mg.
   • After 2 days: m(2) = 450(0.83)^2 ≈ 450(0.6889) ≈ 310.0 mg.
   • After 5 days: m(5) = 450(0.83)^5 ≈ 450(0.3939) ≈ 177.3 mg.
4. Long-term behavior: Because 0.83 < 1, m(t) decreases exponentially and approaches 0 as t becomes large (the radon decays away over time).
5. Half-life (time to reach half the initial mass): Solve (0.83)^t = 0.5. Taking natural logs: t = ln(0.5)/ln(0.83) ≈ 3.72 days. So about 3.7 days are required for the sample to fall to half its original mass.

In summary: the mass decreases exponentially, losing about 17% each day (multiplied by 0.83 per day), with a half-life of approximately 3.72 days and a long-term trend toward zero mass.