You go to the doctor and he gives you 15 milligrams of radioactive dye. After 20 minutes, 7 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 
   minutes at the doctor's office.

To solve this problem, we need to model the radioactive dye's decay over time. This can be done using an exponential decay formula:

\[
A(t) = A_0 e^{-kt}
\]

Where:
- \( A(t) \) is the amount of dye remaining after time \( t \) (in minutes).
- \( A_0 \) is the initial amount of dye (15 milligrams in this case).
- \( k \) is the decay constant.
- \( t \) is the time elapsed (in minutes).

### Step 1: Find the decay constant \( k \)

We are told that 7 milligrams of dye remain after 20 minutes. Using this information, we can substitute the values into the exponential decay formula:

\[
7 = 15 e^{-20k}
\]

Solve for \( k \):

\[
\frac{7}{15} = e^{-20k}
\]

Take the natural logarithm of both sides:

\[
\ln\left(\frac{7}{15}\right) = -20k
\]

\[
k = -\frac{1}{20} \ln\left(\frac{7}{15}\right)
\]

Now, calculate the value of \( k \):

\[
k = -\frac{1}{20} \ln\left(0.4667\right)
\]

\[
k \approx 0.0382
\]

### Step 2: Determine when the dye decreases to 2 milligrams

Next, we need to find the time \( t \) when the amount of dye remaining is 2 milligrams:

\[
2 = 15 e^{-kt}
\]

Solve for \( t \):

\[
\frac{2}{15} = e^{-kt}
\]

Take the natural logarithm of both sides:

\[
\ln\left(\frac{2}{15}\right) = -kt
\]

\[
t = -\frac{1}{k} \ln\left(\frac{2}{15}\right)
\]

Substitute the value of \( k \):

\[
t = -\frac{1}{0.0382} \ln\left(0.1333\right)
\]

\[
t \approx \frac{1}{0.0382} \times 2.015
\]

\[
t \approx 52.75 \text{ minutes}
\]

### Step 3: Total time spent at the doctor's office

Since you were given the dye as soon as you arrived, the total time you spend at the doctor's office is approximately:

\[
t \approx 53 \text{ minutes}
\]

Thus, you will spend **53 minutes** at the doctor's office.

Would you like more details on the steps? Here are some related questions:

1. What is the exponential decay formula and why is it used here?
2. How does the decay constant \( k \) affect the rate of decay?
3. How is the natural logarithm useful in solving exponential equations?
4. What factors affect the time it takes for the dye to reach a safe level?
5. Can we use a similar approach to model other forms of decay, such as population or interest decay?

**Tip**: In exponential decay problems, small changes in the decay constant \( k \) can have a significant impact on the time it takes for a quantity to reduce to a target level.

You go to the doctor and he gives you 15 milligrams of radioactive dye. After 20 minutes, 7 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.

Solve an exponential decay problem involving 15 milligrams of radioactive dye in the body. After 20 minutes, 7 milligrams remain. Learn how to calculate the time needed for the dye to reduce to 2 milligrams using exponential decay and natural logarithms. Suitable for high school students in Grades 11-12.

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