The growth of a bacterial colony can be modeled as a first-order process in which the probability of cell division is linear with respect to time such that

dN/N = ζdt, where dN is the number of cells that divide in the time interval dt, and ζ is a constant.

a. Use the preceding expression to show that the number of cells in the colony is given by N = N₀e^ζt, where N is the number of cells in the colony and N₀ is the number of cells present at t = 0.

b. The generation time is the amount of time it takes for the number of cells to double. Using the answer to

part (a), derive an expression for the generation time.

c. In milk at 37 °C, the bacteria lactobacillus acidophilus has a generation time of about 75 min.

Construct a plot of the acidophilus concentration as a function of time for time intervals of 15, 30, 45, 60, 90, 120, and 150 minutes after a colony of size N₀ is introduced to a container of milk.