# CALC -- integrals!

Iodine-131 is used to treat hypo-thyroidism since it is preferentially absorbed by the thyroid and typically involves a total radiation dosage of 10,000,000 millirem. Iodine-131 has a half-life of eight days.

1. Set up the appropriate integral to represent the radiation dosage delivered by the absorbed iodine in eight days.

2. Use the fact that eight days is the half-life of the isotope to find the initial radiation intensity in millrems/hour.

3. To the nearest 10 millirems, how much of the total radiation is delivered in six weeks?

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1. well, we know the intensity is exponential in time.
I = Io e^-kt
the half life is 8 days
I/Io = .5 = e^-8k
ln .5 = -8k
-.693147=-8k
k = .0866
so
I = Io e^-.0866 t
now total over 8 days, call it D for dose
D = int I dt
D = int dt Io e^-.0866 t from t=0 to t = 8
D = (Io/-.0866) [ e^-.0866(8)-e^0]
e^0 = 1 of course
e^-.6931 = .5 of course
so
D = .5 Io/.0866
or
Io = 5.77 D
D is given of course so you are there in millirems /24 hours
I will leave the doing it in hours and the six weeks thing for you

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