The population of a Seattle suburb is growing at a rate of...

The population of a Seattle suburb is growing at a rate of $3.2 \%$ per year. If 30,000 people lived in the suburb in 2008 , determine how many people will live in the town in 2015. Use $y=y_{0} e^{0.032 t}$

Key Concepts

Time Interval Calculation

The time interval (t) is an essential component in the exponential model, representing the period over which the growth occurs. It is calculated by determining the difference between the final and initial time points, ensuring that the units of time match those used in the growth rate for accurate calculations.

Growth Rate

The growth rate is a constant that indicates how quickly the quantity is increasing over time. Expressed in the exponent of the exponential function, it determines the rapidity of the growth process, with higher rates resulting in faster increases in the quantity.

Continuous Growth Model

The continuous growth model is expressed with the formula y = y? e^(rt), where y? is the initial amount, r is the constant growth rate, and t is the elapsed time. This model is used to represent situations where growth occurs continuously rather than at discrete time intervals, providing a smooth and accurate description of the growth process.

Exponential Growth

Exponential growth describes a process where the quantity increases at a rate proportional to its current value. This behavior is characteristic of many natural and economic processes, such as population growth, where the rate of increase accelerates over time as the quantity gets larger.

Initial Value

The initial value (y?) represents the starting point of the process being modeled. It serves as the baseline from which all subsequent growth is measured, making it a critical factor in calculating future values in exponential growth scenarios.

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