Exponential decay problem?

You go to the doctor and he injects you with 13 milligrams of radioactive dye. After 12 minutes, 4.75 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 doctor will sound the alarm whenever 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 and the amount of dye decays exponentially?

Thank you in advance!

Re: Exponential decay problem?

$\displaystyle y(t) = y_0 e^{kt}$

$\displaystyle y_0 = 13 \, mg$

$\displaystyle y(12) = 4.75 \, mg$

use the value for $\displaystyle y(12)$ to solve for $\displaystyle k$ (you should get a negative value)

$\displaystyle 4.75 = 13e^{k \cdot 12}$

use the value found for $\displaystyle k$ and solve for $\displaystyle t$ in the equation

$\displaystyle 2 = 13e^{kt}$

to get the time in minutes when the dye has decayed to 2 mg