A microbiology lab begins a culture containing 500 bacterial cells at time t=0 hours. Three hours later, the culture is measured to contain 2,000 cells. For the duration of the experiment, the lab technician notes:
“The rate at which the number of bacteria changes is directly proportional to the present number of bacteria.”
Let N(t) denote the number of bacterial cells t hours after the culture is started.
(1) Formulate a differential equation that models the technician’s observation.
(2) Solve the differential equation to obtain the general expression for N(t) in terms of an unknown constant of proportionality.
(3) Determine the numerical value of the proportionality constant k using the data provided.
(4) Using your model, calculate the time (in hours, rounded to three decimal places) required for the culture to reach 10,000 cells.
(5) State the units of k and explain how those units are consistent with the model.