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Mathematical modeling is a process that uses mathematical structures and formulas to represent and solve problems from real-world scenarios. In the realm of finances, modeling can help predict outcomes, such as future costs or investment returns. Take Joe's situation as an example: the problem involves forecasting how much rent he will pay in the future based on a predictable percentage increase each year. To model this, we use an exponential function because the rent increases by a fixed percentage, rather than a fixed amount.

To create an accurate model, we start by defining the variables. Our initial value, or 'a', is the current rent (\$1,400). We then determine 'b', the growth factor. Since the rent is increasing by 1.25% annually, 'b' is 1.0125 when converted from a percentage to a decimal. These values shape the exponential function, which we denote as f(x) where x signifies the number of years. For Joe, the mathematical model takes the form of an equation: \(f(x) = 1400 \times (1.0125)^x\). Using this, we can calculate the future rent at any given point in time.

Percentage increase is a way of expressing the change in a value as a proportion of its original value. In financial matters, such as rent or investment, a percentage increase is commonly used to set rates for growth or inflation. Understanding how to calculate and interpret percentage increases is crucial for making informed decisions. The percentage increase is typically converted into a decimal to perform calculations, which is done by dividing the percentage by 100.

For instance, a 1.25% increase in Joe's rent means every year the rent will be 1.25% higher than the previous year's rent. Mathematically, we represent this as a multiplier (1 + 1.25/100), which equals 1.0125. This multiplier is applied to the current value to find the new value after the increase. When compounded annually, as in Joe's case, the rent is multiplied by this factor every year, which leads us to the concept of exponential growth.

Exponential growth occurs when a quantity increases by a consistent percentage over equal intervals of time. It's characterized by a rapidly increasing value that can be modeled by an exponential function. In finance, understanding exponential growth is key for managing investments, loans, and any situation where compounding occurs, such as interest rates or increasing costs.

In Joe's case, the apartment rent increases by 1.25% each year. This is a perfect example of exponential growth where the previous year's rent is the base for the next year's percentage increase, resulting in a compounding effect. After 12 years, the exponential growth is calculated with the function \(f(x) = 1400 \times (1.0125)^x\), where x=12. Exponentially growing quantities can surprise with their rapid escalation over time, which underscores the importance of early planning and assessment in financial situations.