2 PROJECT C Problem 3: Lets return to our coffee example using our new and improved model. You notice that you feel the affects of the caffeine most after about 40 minutes of drinking the coffee. a. Using your answers to Problem 1 and Problem 2 c., solve for a. b. Now that you have a and b, find k to determine the formulae for A(t) and B(t). c. Using one of the root approximation techniques that we have learned about this semester, find out when B(t) = 30 so you know how long you will have to wait to go to sleep!When a patient takes a drug, it can be quite important to know the amount of that drug in the patient's blood after t hours, B(t). If we assume that the drug is absorbed into the blood instantly and the body uses up some proportion, b, of it every hour, then we know that (1) 3'0) = bB(t) From this we can see that (2) 1100: 1:8 = and integrating both sides gives us that ln(B(1)) = bt + c for some 0. Exponentiating each side and setting k = eC gives us (3) e'bt\": = B(t) = ke'b' b for some positive k. Problem 1: Suppose that you drink a cup of coffee that contains about 150mg of caffeine. Every four hours, the amount in your blood will be cut in half. a. Find 1: and b to determine the formula for B(t). 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 caeine 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 1 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(I) and 8(1) are related like so: A'(t) = aA(t) (4) B'(t) = aA(t) 1913(1) 3(0) 2 0 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