Question

You go to the doctor and he gives you 14 milligrams of dye. After 24 minutes, 7.75 milligrams of the dye remain in your system. To leave dr office, you must go through a radiation detector w/out 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 dr take, assuming you were given the dye as soon as you arrived. Give answer to nearest minute.

You will spend ______ minutes at doctor office.

Answer 1

Assume the radioactive dye decays exponentially. Let t = time in minutes. The amount present obeys the equation

A(t) = A0e-kt,

k = a positive constant

t = minutes elapsed from the time when A was A0

In milligrams,

4 = 15e-k(20)

k = [-ln(4/15) / 20] min-1 ≅ 0.06609 min-1

Set

2 = 15e-0.06609t

and solve for t.

t = -ln(2/15) / (-0.06609) min ≅ 30.49 min

It takes about 30.49 minutes for 15 milligrams to decay down to 2 milligrams.

Explanation: