Suppose 4 mg of a drug is injected into a person's bloodstream. as the drug is metabolized, the quantity diminishes at the continuous rate of 3% per hour?

Let Q(t) = the mass (mg) remaining after t hours.
We are told that Q diminishes by 3% every hour.

When t = 0, Q = 4 mg
When t = 1,  Q = 4*0.97 mg
When t = 2, Q = 4*(0.97)² mg

By induction,


Q'(t) = 0.97Q
Therefore the rate of decrease is 3% per hour.

The person receives an additional dosage when Q falls to 0.50 mg. This happens when


After the second injection, the mass is now 4.5 mg. Therefore


When the mass again reaches 0.50 mg, then


Answers: 
(a) 
(b) 3% per hour
(c) 68.3 hours
(d) 72.1 hours

The answer would be 27 because assuming that each package has 1 bar of soap and 1 tube of toothpaste, after 27 packages they would run out of toothpaste