A thyroid cancer patient is given a dosage of 131I

To find out what fraction of the dosage of 131I will still be in the patient's thyroid after 32.4 days, we can use the exponential decay formula:

N(t) = N0 * (1/2)^(t/T1/2)

Where:
N(t) = final amount of 131I in the patient's thyroid after time t
N0 = initial dosage of 131I given to the patient
t = time elapsed (32.4 days)
T1/2 = half-life of 131I (8.1 days)

Plugging in the values:
N(32.4) = N0 * (1/2)^(32.4/8.1)
N(32.4) = N0 * (1/2)^4

N(32.4) = N0 * (1/16)

After 32.4 days, only 1/16th of the initial dosage of 131I will still be in the patient's thyroid.