👉 When you solve a problem involving the application of multiple mathematical concepts, such as combining exponential growth and decay, you typically end up with an expression that describes a specific outcome over time. For instance, if you're dealing with a population growing exponentially and then experiencing a sudden decline due to a disease, the final expression might be a product of an exponential growth function (representing the initial increase) and a decay function (accounting for the decrease). This results in a complex function that captures both the accelerating and then decelerating phases of the population change, often expressed as \( P(t) = P_0 e^{rt} (1 - k) \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate, \( k \) is the decay rate, and \( e \) is the base of natural logarithms. This expression encapsulates the initial exponential growth, the impact of a sudden reduction (modeled by \( k \)), and how these factors evolve over time.
