The concentration M(t) (in mg\,L−1) of a medication in a patient’s bloodstream satisfies dtdM=kM for t≥0, where t is measured in hours. Immediately after administration, M(0)=250. Four hours later, M(4)=180.
(a) Determine an exact expression for the constant k.
(b) Compute the time, to the nearest minute, at which the concentration will first fall to 50\,mg\,L−1.
(c) If a second identical dose is administered precisely at the time found in part (b), write an expression for the total concentration immediately after the second dose. State whether the growth constant k changes and explain your reasoning.