Q.The rate of growth of the population of a city at any time $t$ is proportional to the size of the population. For a certain city the constant of proportionality is 0.04. Find the population of the city after 25 years if the initial population is 10,000. [Take $e=2.7182$]

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) t …