Involve exponential growth. Suppose a bacterial culture...

Involve exponential growth. Suppose a bacterial culture initially has 400 cells. After 1 hour, the population has increased to $800 .$ Find an equation for the population at any time. What will the population be after 10 hours?

Key Concepts

Exponential Growth

Exponential growth refers to an increase in a quantity where the rate of growth is proportional to the current amount. This results in the quantity increasing by a constant ratio over equal increments of time, leading to rapid escalation. In many real-world applications, such as population dynamics, this concept allows for the modeling of systems where growth compounds continuously or in discrete intervals.

Exponential Function Modeling

Exponential function modeling involves representing a quantity that grows or decays by an exponential function, typically in the form P(t) = P? · b^(t) or P(t) = P? · e^(rt). Here, P? is the initial quantity, b is the growth (or decay) factor over one time period, and r is the continuous growth rate. This model is essential for predicting future values based on known starting conditions and growth characteristics.

Determining Growth Factors

Determining the growth factor involves understanding how much a quantity increases over a given time period. When given an interval where the quantity doubles or changes by another specific factor, one can calculate the base of the exponential function. This calculation is crucial for accurately setting up the exponential model and making predictions about future behavior.

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