a. Find k and b to determine the formula for 3(1). b. Verify that your answer to a. actually satises (1). c. Most people will not be able to sleep with more than about 30mg of caffeine in their system. How long will it take until there is only 30mg of caffeine in your blood? The model above is only accurate if the drug administered takes an insignicant amount of time to enter the blood stream. Inhalers and injections usually fall into this category; however, drugs taken orally take a signicant amount of time to be absorbed. In that case we model the situation differently First, we say that the amount of the drug in the digestive tract after t hours is A(t), and that a proportion, a, of this is absorbed into the blood. Then we can put this together with our simpler model to see that A(t) and 30') are related like so: A'(t) = aA(t) (4) B'(z) = aA(t) bB(t) 3(0) = 0 Because the top line of (4) looks exactly like (1), we know thatA (t) = ke'\" without having to do any new work. On the other hand, solving for B(t) is acutally quite involved so if you want to know how to do that take MTH350! Here is the solution: ak at_ bt ba(e e ) (5) BO) = Problem 2: Lets explore this situation a little more' a. Verify that our formula for A(t) satises the top line of (4). b. Verify that our formula for 30') satises the bottom two lines of (4). c. Find the time at which B(t) is at its highest