Q.In a certain culture of bacteria the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours.

When a population (of people, bacteria, or any self-reproducing quantity) grows at a rate proportional to its own current size, that statement translates directly into the differential equation dP/dt = kP with k > 0, which separates and integrates to the exponential law of natural growth P = c·e^(kt), where c is the value of P at the reference time t = 0. In word problems, two pieces of numerical data — usually the size at t = 0 and the size at one later instant — are used to pin down both c and k (or, more efficiently, to find the RATIO by which the population changes over a given interval directly, without ever computing the decimal value of k, by raising e^(kt₁) to the power t₂/t₁). This model underlies every 'doubles in n years', 'increases from A to B in n years', or 'bacteria multiply' word problem in the chapter, and it is mathematically the mirror image of the radioactive-decay model, differing only in the sign of the proportionality constant.