QuestionYou go to the doctor and he gives you 15 milligrams of radioactive dye. After 12 minutes, 8 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.
You will spend $\square$ minutes at the doctor's office.

Studdy Solution
We need to find the time $t$ when $A(t) = 2$ milligrams. Substitute into the decay model:
$$2 = 15 e^{-kt}$$
Solve for $t$:
$$\frac{2}{15} = e^{-kt}$$
Take the natural logarithm of both sides:
$$\ln\left(\frac{2}{15}\right) = -kt$$
Substitute the value of $k$ from Step 2:
$$t = -\frac{1}{k} \ln\left(\frac{2}{15}\right)$$
$$t = \frac{12}{\ln\left(\frac{8}{15}\right)} \ln\left(\frac{2}{15}\right)$$
Calculate $t$ using a calculator:
$$t \approx 28.4$$
Round to the nearest minute:
$$t \approx 28$$
You will spend $\boxed{28}$ minutes at the doctor's office.