1. a patient is in the hospital receiving adrug via intravenous (IV) therapy at a ratei = i(t) [mg/s]. the drug clears from theblood, owing to the metabolism, with first-order

kinetics and rate constant k [1/s.given that V [L] is the volume of blood inthe body, a differential equation model forthe concentration of the drug in the blood-stream, c = c(t) [mg/L], is: V \frac{d c}{d t}=i-k c V (a) (1 point) to be effective, yet also nontoxic, the drug is required to be maintainedat a steady state concentration of ē [mg/L]. write an expression for the (constant)rate at which the drug should be continuously administered, i [g/s], to maintainthis steady state concentration č. (b) (1 point) write eqn. 1 in terms of the deviation variables: c^{*}(t):=c(t)-\bar{c} i^{*}(t):=i(t)-\bar{i} (c) (3 points) derive the transfer function G(s) := C*(s)/I*(s) that tells us how the concentration of the drug in the blood, c, changes in response to changes in the rate at which the drug is administered, i. here, C*(s) := L[c* (t)] and I*(s) := L[i*(t)] (d) (1 point) write the transfer function in standard gain/time constant form. what is the gain and the time constant of this first-order transfer function? (e) on top of the IV therapy, an extra dose of m [g] of the drug is given via an injection(Dirac delta function input) at time t = 0. i. (2 points) derive the response c(t) (not in deviation form). ii. (2 points) what is the peak concentration of the drug, max; c(t), and when does it occur, arg max; c(t)? iii. (1 point) how much time elapses before the peak decays to 10% of its maximal value?