For this problem, give all numerical answers correct to at least three decimal places The body regulates the level of a certain hormone. If the concentration of the hormone is above its baseline level, the liver metabolizes the hormone, reducing the excess concentration at a rate that is proportional to the excess concentration. The constant of proportionality in the relationship between the elimination rate and the excess concentration is known as the velocity constant of elimination. Its value is positive and is denoted by k (measured in units of hour-1). Denote the excess concentration, measured in mg/L, at time t by y(t). (a) Write down the right side of the differential equation satisfied by y. (Your answer should be in terms of y.) dy = dt (Aside: Can you explain why you have a minus sign in your differential equation?) (b) Tests reveal that the velocity constant of elimination for a particular patient is 0.25 hour" and that their baseline hormone level is too low. A doctor tries to increase the level of hormone by giving the patient a single shot of a hormone supplement at time t = 0. The shot causes the excess hormone concentration to immediately jump to 4.2 mg/L. She will give another shot when the hormone level falls back to 0.6 mg/L above the patient's baseline. After how long will she have to give the next shot? After hours (c) The doctor has another patient whose baseline level is too low. She gives them a hormone supplement shot at time t = 0. This causes their excess hormone concentration to immediately jump to 4.5 mg/L. Then level then falls, due to the hormone being metabolized. Monitoring the excess hormone concentration, the doctor sees that the initial rate of this fall equals 0.7 mg/L/hour. What is this patient's velocity constant of elimination? hour-1 How long would it take the excess level to fall to 0.6 mg/L? | hours Another doctor has a patient whose baseline hormone level is too low. Rather than having to give a series of single shots, he decides to give the patient a continuous intravenous infusion of the hormone supplement. If there were no metabolism, the infusion would cause the concentration of hormone to increase at a constant rate of A mg/L/hr. The velocity constant of elimination for this patient is equal to k hour-1. At the initial time, t = 0, the patient's hormone concentration is at their baseline. (d) Write down the right side of the differential equation satisfied by y. (Your answer should be in terms of y.) dy = dt (e) This differential equation can be rewritten in the more familiar form of = k(M - y) for a particular choice of M. Write down an expression for M in terms of A and k. M= (f) What is the patient's initial excess hormone concentration? y(0) = (g) Using the solution of this differential equation, the excess hormone concentration is given by y(t) = M(1 - ekt). In terms of A and k, what value does the excess hormone concentration approach in the long run? mg/L (h) After doing some tests, the patient's velocity constant of elimination is found to be 0.028 hour"1. The doctor plans to administer the infusion for 36 hours and wants to raise the excess hormone concentration to 3.8 mg/L at that time. What infusion rate must the doctor choose to reach this target? mg/L/hr At this infusion rate, what will the excess hormone concentration be at the half-way time? mg/L At this infusion rate, how long will it take for the excess hormone concentration to reach half of the target level? hours