You go to the doctor and he gives you 15 milligrams of radioactive dye...?

Question

You go to the doctor and he gives you 15 milligrams of radioactive dye. After 24 minutes, 5.25 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?

Answer 1

The amount of the radio-active dye can be modeled by the equation A(t) = A₀e^-kt, 

Let A(t) be in mg and t be in minutes

A₀=A(0)= 15

A(24) =5.25=15e^-k24 =>ln(5.25) =ln(15) -k24 [ taking ln on both sides]

 => k =ln(15/5.25) / 24 = 0.04374

Time t=T for the amount to be 2mg is given by 2=15e^-kT

=>T=ln(15/2)/k =46.06

The visit will take 46 minutes (rounded to minutes).