You go to the doctor and he gives you 10 milligrams of radioactive dye. After 20 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? Give your answer to the nearest minute. You will spend minutes at the doctor's office.

Step 1 ：This problem is about radioactive decay, which follows an exponential decay model. The formula for exponential decay is: \(N(t) = N0 * e^{-λt}\) where: \(N(t)\) is the quantity at time t, \(N0\) is the initial quantity, \(λ\) is the decay constant, and \(t\) is the time.

Step 2 ：We are given that the initial quantity \(N0\) is 10 milligrams and after 20 minutes, the quantity \(N(t)\) is 5.25 milligrams. We can use these values to solve for the decay constant \(λ\).

Step 3 ：Once we have \(λ\), we can solve for the time \(t\) when the quantity \(N(t)\) is 2 milligrams.

Step 4 ：The calculated time it takes for the quantity of dye to reduce to 2 milligrams is approximately 49.95 minutes. This is the time from when the dye was administered.

Step 5 ：However, the question asks for the total time spent at the doctor's office, which includes the initial 20 minutes. Therefore, we need to add these 20 minutes to our calculated time.

Step 6 ：The total time spent at the doctor's office is approximately 69.95 minutes.

Step 7 ：Rounding to the nearest minute, the final answer is: \(\boxed{70}\) minutes.