Suppose that d P / d t=0.15 P(t) represents a mathematical...

Suppose that $d P / d t=0.15 P(t)$ represents a mathematical model for the growth of a certain cell culture, where $P(t)$ is the size of the culture (measured in millions of cells) at time $t > 0$ (measured in hours). How fast is the culture growing at the time when the size of the culture reaches 2 million cells?

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Key Concepts

Exponential Growth

Exponential growth describes processes in which the rate of change of a quantity is directly proportional to the current amount. This leads to a mathematical model where the solution can be expressed in the form P(t) = P?e^(kt), meaning that the population increases at a rate that accelerates over time.

Differential Equations

Differential equations are equations that relate a function to its derivatives. In the context of population dynamics, they provide a way to model how a population changes over time. The equation dP/dt = 0.15 P(t) is a simple first-order linear differential equation that captures the essence of exponential growth.

Rate of Change

The rate of change represents how quickly a quantity is increasing or decreasing at a particular time. In this model, the derivative dP/dt gives the instantaneous growth rate of the cell culture. By substituting the current population size into the equation, one can determine exactly how fast the culture is growing at that moment.

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