How to get the answer to this problem? What's the answer?

Question

You go to the doctor and he gives you 16.8 milligrams of radioactive dye. After 19.00 minutes, 4.00 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.

Answer 1

This is essentially a problem dealing with half life and fraction of drug remaining. The assumption is that decay of the dye/drug is 1st order and so we can use the formula... FR = 0.5ⁿ

FR = fraction remaining

n = number of half lives that have elapsed

From the first part of the question, we can figure out the half life of the dye as follows:

FR = fraction remaining = 4.00 mg / 16.8 mg = 0.2381 and this is after 19 minutes

FR = 0.5ⁿ = 0.2381 = 0.5ⁿ

log 0.2381 = log 0.5 n

-0.623 = -0.301 n

n = 2.07 half lives have elapsed and this is equal to 19 minutes

19.00 min / 2.07 half lives = 9.18 minutes per half life

Now that we have the half life, we can solve for the time it will take for the dye to decay to 2 mg:

2 mg / 16.8 mg = 0.119 which would be the fraction remaining when 2 mg are left

FR = 0.119 = 0.5ⁿ

log 0.119 = log 0.5 n

n = 3.07 half lives

3.07 half lives x 9.18 minutes / half life = 28 minutes