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Population growth is an essential area of study in demographics and environmental science. In a city, population growth can significantly impact resources, infrastructure, and the environment. When discussing population growth, we refer to how the number of individuals in a population increases over time.

In this case, we have a linear growth model, meaning the population increases by a constant number each year. Here, the city's population increases by 850 people annually. Such a model is simplified compared to real-world scenarios where factors like birth rates, death rates, immigration, and policy changes could influence growth. However, linear models provide a clear, initial understanding of growth trends.

With an initial population of 30,700 in 2010 and a steady increase, predictions become straightforward. By applying this model, we can anticipate how population changes over a given timeframe.

Applied calculus involves using mathematical concepts and techniques to solve real-world problems. In population growth modeling, calculus may be used to understand changes and dynamics over continuous time.

In this exercise, however, the growth is simplified to a linear form, which doesn't typically require advanced calculus, but instead uses basic algebraic functions. The formula for population growth, \(P(t) = 30,700 + 850t\), is an application of simple linear equations. Here, \(P(t)\) represents the population depending on \(t\), the number of years since 2010.

In more complex scenarios, calculus can help analyze non-linear growth, where factors influencing populations change dynamically. Calculus allows us to find rates of change at any given moment and model these changes accurately, which is invaluable in more sophisticated population studies.

Mathematical modeling is a process of representing real-world situations using mathematical language and structures. It provides a way to interpret and predict behaviors and outcomes using mathematical expressions.

For this exercise, the linear function \(P(t) = 30,700 + 850t\) is the model representing the city's population over time. This model helps to predict future population sizes and understand how changes happen over time.

Models are invaluable because they transform abstract concepts into concrete, manageable forms that can be analyzed and tested. The linear model's simplicity allows for easy calculations and predictions. However, it's crucial to remember that all models are simplifications. Despite being simple, this model effectively shows basic growth trends and offers a starting point for more detailed analysis, should additional data become available.