A doctor prescribes 375 milligrams of a therapeutic drug that decays by about $35 \%$ each hour.
Write an exponential model representing the amount of the drug remaining in the patient's system after $t$ hours. Find the amount of the drug that would remain in the patient's system after 3 hours. Round to the nearest milligram.

Step 1 ：The problem is asking for an exponential model, which is a mathematical model that describes a pattern of exponential growth or decay. In this case, the drug decays by about 35% each hour, which means that each hour, the amount of the drug in the patient's system is 65% of the amount from the previous hour. This can be represented by the exponential decay model \(A = P(1 - r)^t\), where \(A\) is the amount of the drug remaining after \(t\) hours, \(P\) is the initial amount of the drug, \(r\) is the rate of decay, and \(t\) is the time in hours.

Step 2 ：In this case, \(P = 375\) milligrams, \(r = 0.35\), and we want to find \(A\) when \(t = 3\) hours.

Step 3 ：Substitute the given values into the formula: \(A = 375(1 - 0.35)^3\)

Step 4 ：Solve the equation to find the amount of the drug remaining in the patient's system after 3 hours. Round the result to the nearest milligram.

Step 5 ：Final Answer: The amount of the drug that would remain in the patient's system after 3 hours is \(\boxed{103}\) milligrams.