Modeling Population Growth with Exponential Functions

Write an exponential function to model each situation and find the amount after the specified time. For example, given a population of 1,236,000 that grows at an interest rate of 1.3% over 10 years, find the final population.

Exponential functions are a powerful tool in modeling processes that change at a constant percentage rate, such as population growth or compound interest. The problem presents a classic example of this application. When addressing such a problem, the key is to understand the basic form of an exponential function, which can be expressed generally as initial amount times the base of the natural logarithm raised to the power of the growth rate times time. Here, the base of the exponential function reflects the multiplicative rate of change, which is crucial for accurately modeling real-world phenomena.

When tackling this type of question, consider not only the immediate figures provided, such as initial population and growth rate, but also how these parameters influence the model over time. The understanding of continuous versus discrete growth models is fundamental, although in this problem, the focus remains on continuous growth given the constant percentage rise per time unit. Deciphering these relationships is essential to mastering the application of exponential functions in various contexts, which is a pivotal learning outcome in Discrete Mathematics and related fields. Such exercises serve to develop skills needed to formulate and solve problems where change occurs exponentially.