Consider a biochemical reaction in which a certain substance is both produced and consumed. The concentration of this substance at time t is defined to be c(t). Assume that the function c obeys the following differential equation:
dc/dt = Kmax c/(k + c) รขยย rc
Where Kmax, k, and r are all positive constants. The first term on the right hand side of this equation denotes the concentration-dependent production and the second denotes the consumption.
(a) What is the maximal rate at which the substance is produced?
(b) At what concentration is the production rate 50% of this maximum value?
(c) If the production is turned off, the substance decays. How long will it take for the concentration to drop by 50%?
(d) At what concentration does production balance consumption?