You oversee shipping of radioactive imaging markers for hospitals. A hospital needs $4 \mathrm{mg}$ of a particular marker for a test to be done in 4 days. After filling the order, it will take three days for the dye to arrive. The dye has a half-life of 42 hours. How many milligrams of the dye need to be shipped so the hospital will have the needed 4 $\mathrm{mg}$ in three days?
Total milligrams of dye that need to be shipped is

Step 1 ：The hospital needs the dye in 4 days, but it will take 3 days for the dye to arrive. This means that the dye needs to last for a total of 7 days, or 168 hours.

Step 2 ：The half-life of a substance is the time it takes for half of the substance to decay. In this case, the half-life of the dye is 42 hours. This means that after 42 hours, half of the dye will have decayed.

Step 3 ：We can calculate the amount of dye needed by using the formula for exponential decay, which is: \(N = N_0 * (1/2)^(t/T)\) where: \(N\) is the final amount of the substance, \(N_0\) is the initial amount of the substance, \(t\) is the time that has passed, \(T\) is the half-life of the substance.

Step 4 ：In this case, we know that \(N = 4\), \(t = 168\), and \(T = 42\). We can rearrange the formula to solve for \(N_0\): \(N_0 = N / (1/2)^(t/T)\)

Step 5 ：We can now plug in the values and calculate \(N_0\).

Step 6 ：Final Answer: The hospital needs to be shipped \(\boxed{64}\) milligrams of the dye.